:: MFOLD_2 semantic presentation

REAL is non empty non trivial non finite set
NAT is non empty epsilon-transitive epsilon-connected ordinal Element of bool REAL
bool REAL is non empty set
omega is non empty epsilon-transitive epsilon-connected ordinal set
bool omega is non empty set
bool NAT is non empty set
COMPLEX is non empty non trivial non finite set
RAT is non empty non trivial non finite set
INT is non empty non trivial non finite set
[:REAL,REAL:] is Relation-like non empty set
bool [:REAL,REAL:] is non empty set
K281() is non empty V72() L8()
the carrier of K281() is non empty set
K286() is non empty V72() V94() V95() V96() V98() V148() V149() V150() V151() V152() V153() L8()
K287() is non empty V72() V96() V98() V151() V152() V153() M13(K286())
K288() is non empty V72() V94() V96() V98() V151() V152() V153() V154() M16(K286(),K287())
K290() is non empty V72() V94() V96() V98() L8()
K291() is non empty V72() V94() V96() V98() V154() M13(K290())
1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
[:1,1:] is Relation-like non empty set
bool [:1,1:] is non empty set
[:[:1,1:],1:] is Relation-like non empty set
bool [:[:1,1:],1:] is non empty set
[:[:1,1:],REAL:] is Relation-like non empty set
bool [:[:1,1:],REAL:] is non empty set
[:[:REAL,REAL:],REAL:] is Relation-like non empty set
bool [:[:REAL,REAL:],REAL:] is non empty set
2 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
[:2,2:] is Relation-like non empty set
[:[:2,2:],REAL:] is Relation-like non empty set
bool [:[:2,2:],REAL:] is non empty set
TOP-REAL 2 is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional 2 -locally_euclidean 2 -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL 2) is functional non empty set
K463() is V226() TopStruct
the carrier of K463() is V128() set
RealSpace is non empty strict Reflexive discerning V177() triangle Discerning V226() MetrStruct
R^1 is non empty strict TopSpace-like V226() TopStruct
K470() is TopSpace-like V226() SubSpace of R^1
K448(K470(),K470()) is strict TopSpace-like TopStruct
the carrier of K448(K470(),K470()) is set
K472() is non empty strict TopSpace-like V226() V228() SubSpace of R^1
the carrier of K472() is non empty V128() set
bool the carrier of K472() is non empty set
bool (bool the carrier of K472()) is non empty set
Tunit_circle 2 is non empty TopSpace-like T_0 T_1 T_2 V229() SubSpace of TOP-REAL 2
the carrier of (Tunit_circle 2) is non empty set
[: the carrier of K472(), the carrier of (Tunit_circle 2):] is Relation-like non empty set
bool [: the carrier of K472(), the carrier of (Tunit_circle 2):] is non empty set
CircleMap is Relation-like the carrier of K472() -defined the carrier of K472() -defined the carrier of (Tunit_circle 2) -valued the carrier of (Tunit_circle 2) -valued Function-like non empty total total quasi_total quasi_total onto continuous Element of bool [: the carrier of K472(), the carrier of (Tunit_circle 2):]
c[10] is Element of the carrier of (Tunit_circle 2)
Topen_unit_circle c[10] is non empty strict TopSpace-like T_0 T_1 T_2 V240( Tunit_circle 2) SubSpace of Tunit_circle 2
the carrier of (Topen_unit_circle c[10]) is non empty set
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural real finite finite-yielding V36() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding V117() V118() V119() R-orthogonal R-normal R-orthonormal Element of NAT
K386(0,1) is non empty Element of bool REAL
R^1 K386(0,1) is non empty Element of bool the carrier of K472()
K472() | (R^1 K386(0,1)) is non empty strict TopSpace-like V226() SubSpace of K472()
the carrier of (K472() | (R^1 K386(0,1))) is non empty V128() set
[: the carrier of (Topen_unit_circle c[10]), the carrier of (K472() | (R^1 K386(0,1))):] is Relation-like non empty complex-yielding V117() V118() set
bool [: the carrier of (Topen_unit_circle c[10]), the carrier of (K472() | (R^1 K386(0,1))):] is non empty set
c[-10] is Element of the carrier of (Tunit_circle 2)
Topen_unit_circle c[-10] is non empty strict TopSpace-like T_0 T_1 T_2 V240( Tunit_circle 2) SubSpace of Tunit_circle 2
the carrier of (Topen_unit_circle c[-10]) is non empty set
1 / 2 is non empty V21() real ext-real positive non negative Element of REAL
3 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
3 / 2 is non empty V21() real ext-real positive non negative Element of REAL
K386((1 / 2),(3 / 2)) is non empty Element of bool REAL
R^1 K386((1 / 2),(3 / 2)) is non empty Element of bool the carrier of K472()
K472() | (R^1 K386((1 / 2),(3 / 2))) is non empty strict TopSpace-like V226() SubSpace of K472()
the carrier of (K472() | (R^1 K386((1 / 2),(3 / 2)))) is non empty V128() set
[: the carrier of (Topen_unit_circle c[-10]), the carrier of (K472() | (R^1 K386((1 / 2),(3 / 2)))):] is Relation-like non empty complex-yielding V117() V118() set
bool [: the carrier of (Topen_unit_circle c[-10]), the carrier of (K472() | (R^1 K386((1 / 2),(3 / 2)))):] is non empty set
[:COMPLEX,COMPLEX:] is Relation-like non empty set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like non empty set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:RAT,RAT:] is Relation-like non empty set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is Relation-like non empty set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is Relation-like non empty set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is Relation-like non empty set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is Relation-like non empty set
[:[:NAT,NAT:],NAT:] is Relation-like non empty set
bool [:[:NAT,NAT:],NAT:] is non empty set
F_Real is non empty non degenerated non trivial V70() almost_left_invertible V93() V94() V96() V98() right-distributive left-distributive right_unital well-unital V104() left_unital Abelian add-associative right_zeroed L11()
addreal is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [:[:REAL,REAL:],REAL:]
multreal is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [:[:REAL,REAL:],REAL:]
G11(REAL,addreal,multreal,1,0) is V93() L11()
the carrier of F_Real is non empty non trivial V128() set
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural real finite finite-yielding V36() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding V117() V118() V119() R-orthogonal R-normal R-orthonormal set
{{},1} is non empty finite set
the carrier of F_Real * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Real
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
REAL 0 is functional non empty FinSequence-membered FinSequenceSet of REAL
0 -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
- 1 is non empty V21() real ext-real non positive negative Element of REAL
Seg 1 is non empty finite 1 -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
{1} is non empty finite set
Seg 2 is non empty finite 2 -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
{1,2} is non empty finite set
sqrt 1 is V21() real ext-real Element of REAL
real_dist is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [:[:REAL,REAL:],REAL:]
MetrStruct(# REAL,real_dist #) is strict MetrStruct
TopSpaceMetr RealSpace is metrizable TopStruct
the carrier of R^1 is non empty V128() set
rng {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty trivial proper V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural real finite finite-yielding V36() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding V117() V118() V119() increasing V121() V122() V123() V126() V127() V128() V129() V131() with_non-empty_elements R-orthogonal R-normal R-orthonormal Element of bool REAL
n is Relation-like Function-like set
M is set
M |` n is Relation-like Function-like set
dom (M |` n) is set
n " M is set
p is set
n1 is set
[p,n1] is set
{p,n1} is non empty finite set
{p} is non empty finite set
{{p,n1},{p}} is non empty finite V36() set
n1 is set
[p,n1] is set
{p,n1} is non empty finite set
{{p,n1},{p}} is non empty finite V36() set
n is Relation-like Function-like set
p is set
M is set
M |` n is Relation-like Function-like set
(M |` n) " p is set
n " p is set
n1 is set
p1 is set
[n1,p1] is set
{n1,p1} is non empty finite set
{n1} is non empty finite set
{{n1,p1},{n1}} is non empty finite V36() set
p1 is set
[n1,p1] is set
{n1,p1} is non empty finite set
{{n1,p1},{n1}} is non empty finite V36() set
n is TopStruct
the carrier of n is set
M is TopStruct
the carrier of M is set
[: the carrier of n, the carrier of M:] is Relation-like set
bool [: the carrier of n, the carrier of M:] is non empty set
p is Relation-like the carrier of n -defined the carrier of M -valued Function-like quasi_total Element of bool [: the carrier of n, the carrier of M:]
p /" is Relation-like the carrier of M -defined the carrier of n -valued Function-like quasi_total Element of bool [: the carrier of M, the carrier of n:]
[: the carrier of M, the carrier of n:] is Relation-like set
bool [: the carrier of M, the carrier of n:] is non empty set
dom p is Element of bool the carrier of n
bool the carrier of n is non empty set
[#] n is non proper Element of bool the carrier of n
rng p is Element of bool the carrier of M
bool the carrier of M is non empty set
[#] M is non proper Element of bool the carrier of M
[:{},{}:] is Relation-like finite set
bool [:{},{}:] is non empty finite V36() set
n1 is Relation-like non-empty empty-yielding {} -defined {} -valued Function-like one-to-one constant functional empty onto bijective V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural real finite finite-yielding V36() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding V117() V118() V119() R-orthogonal R-normal R-orthonormal Element of bool [:{},{}:]
n1 " is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural real finite finite-yielding V36() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding V117() V118() V119() R-orthogonal R-normal R-orthonormal Element of REAL
dom (p /") is Element of bool the carrier of M
rng (p /") is Element of bool the carrier of n
(p /") /" is Relation-like the carrier of n -defined the carrier of M -valued Function-like quasi_total Element of bool [: the carrier of n, the carrier of M:]
p1 is Relation-like non-empty empty-yielding {} -defined {} -valued Function-like one-to-one constant functional empty onto bijective V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural real finite finite-yielding V36() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding V117() V118() V119() R-orthogonal R-normal R-orthonormal Element of bool [:{},{}:]
p1 " is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural real finite finite-yielding V36() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding V117() V118() V119() R-orthogonal R-normal R-orthonormal Element of REAL
p is TopStruct
n1 is TopStruct
the carrier of p is set
the carrier of n1 is set
[: the carrier of p, the carrier of n1:] is Relation-like set
bool [: the carrier of p, the carrier of n1:] is non empty set
p1 is Relation-like the carrier of p -defined the carrier of n1 -valued Function-like quasi_total Element of bool [: the carrier of p, the carrier of n1:]
p1 /" is Relation-like the carrier of n1 -defined the carrier of p -valued Function-like quasi_total Element of bool [: the carrier of n1, the carrier of p:]
[: the carrier of n1, the carrier of p:] is Relation-like set
bool [: the carrier of n1, the carrier of p:] is non empty set
n is TopSpace-like TopStruct
the carrier of n is set
M is TopSpace-like TopStruct
the carrier of M is set
bool the carrier of M is non empty set
[: the carrier of n, the carrier of M:] is Relation-like set
bool [: the carrier of n, the carrier of M:] is non empty set
p is Element of bool the carrier of M
M | p is strict TopSpace-like SubSpace of M
the carrier of (M | p) is set
n1 is Relation-like the carrier of n -defined the carrier of M -valued Function-like quasi_total Element of bool [: the carrier of n, the carrier of M:]
n1 " p is Element of bool the carrier of n
bool the carrier of n is non empty set
n | (n1 " p) is strict TopSpace-like SubSpace of n
the carrier of (n | (n1 " p)) is set
[: the carrier of (n | (n1 " p)), the carrier of (M | p):] is Relation-like set
bool [: the carrier of (n | (n1 " p)), the carrier of (M | p):] is non empty set
p |` n1 is Relation-like the carrier of n -defined the carrier of M -valued p -valued the carrier of M -valued Function-like Element of bool [: the carrier of n, the carrier of M:]
dom n1 is Element of bool the carrier of n
[#] n is non proper open closed Element of bool the carrier of n
rng n1 is Element of bool the carrier of M
[#] M is non proper open closed Element of bool the carrier of M
A is Relation-like the carrier of (n | (n1 " p)) -defined the carrier of (M | p) -valued Function-like quasi_total Element of bool [: the carrier of (n | (n1 " p)), the carrier of (M | p):]
rng A is Element of bool the carrier of (M | p)
bool the carrier of (M | p) is non empty set
dom A is Element of bool the carrier of (n | (n1 " p))
bool the carrier of (n | (n1 " p)) is non empty set
[#] (n | (n1 " p)) is non proper open closed Element of bool the carrier of (n | (n1 " p))
[#] (M | p) is non proper open closed Element of bool the carrier of (M | p)
m is Element of bool the carrier of (M | p)
A " m is Element of bool the carrier of (n | (n1 " p))
S is Element of bool the carrier of M
S /\ p is Element of bool the carrier of M
n1 " S is Element of bool the carrier of n
n1 " m is Element of bool the carrier of n
(n1 " S) /\ the carrier of (n | (n1 " p)) is Element of bool the carrier of n
n1 /" is Relation-like the carrier of M -defined the carrier of n -valued Function-like quasi_total Element of bool [: the carrier of M, the carrier of n:]
[: the carrier of M, the carrier of n:] is Relation-like set
bool [: the carrier of M, the carrier of n:] is non empty set
A /" is Relation-like the carrier of (M | p) -defined the carrier of (n | (n1 " p)) -valued Function-like quasi_total Element of bool [: the carrier of (M | p), the carrier of (n | (n1 " p)):]
[: the carrier of (M | p), the carrier of (n | (n1 " p)):] is Relation-like set
bool [: the carrier of (M | p), the carrier of (n | (n1 " p)):] is non empty set
m is Element of bool the carrier of (n | (n1 " p))
(A /") " m is Element of bool the carrier of (M | p)
S is Element of bool the carrier of n
S /\ (n1 " p) is Element of bool the carrier of n
(n1 /") " S is Element of bool the carrier of M
n1 .: (n1 " p) is Element of bool the carrier of M
(n1 /") " (n1 " p) is Element of bool the carrier of M
(p |` n1) .: m is Element of bool the carrier of M
n1 .: m is Element of bool the carrier of M
(n1 .: m) /\ the carrier of (M | p) is Element of bool the carrier of M
(n1 /") " (S /\ (n1 " p)) is Element of bool the carrier of M
((n1 /") " (S /\ (n1 " p))) /\ the carrier of (M | p) is Element of bool the carrier of M
((n1 /") " S) /\ ((n1 /") " (n1 " p)) is Element of bool the carrier of M
(((n1 /") " S) /\ ((n1 /") " (n1 " p))) /\ the carrier of (M | p) is Element of bool the carrier of M
((n1 /") " (n1 " p)) /\ the carrier of (M | p) is Element of bool the carrier of M
((n1 /") " S) /\ (((n1 /") " (n1 " p)) /\ the carrier of (M | p)) is Element of bool the carrier of M
((n1 /") " S) /\ the carrier of (M | p) is Element of bool the carrier of M
n is TopSpace-like TopStruct
the carrier of n is set
M is TopSpace-like TopStruct
the carrier of M is set
bool the carrier of M is non empty set
[: the carrier of n, the carrier of M:] is Relation-like set
bool [: the carrier of n, the carrier of M:] is non empty set
p is Element of bool the carrier of M
n1 is Relation-like the carrier of n -defined the carrier of M -valued Function-like quasi_total Element of bool [: the carrier of n, the carrier of M:]
n1 " p is Element of bool the carrier of n
bool the carrier of n is non empty set
p |` n1 is Relation-like the carrier of n -defined the carrier of M -valued p -valued the carrier of M -valued Function-like Element of bool [: the carrier of n, the carrier of M:]
dom (p |` n1) is Element of bool the carrier of n
n | (n1 " p) is strict TopSpace-like SubSpace of n
[#] (n | (n1 " p)) is non proper open closed Element of bool the carrier of (n | (n1 " p))
the carrier of (n | (n1 " p)) is set
bool the carrier of (n | (n1 " p)) is non empty set
rng n1 is Element of bool the carrier of M
[#] M is non proper open closed Element of bool the carrier of M
rng (p |` n1) is Element of bool p
bool p is non empty set
M | p is strict TopSpace-like SubSpace of M
[#] (M | p) is non proper open closed Element of bool the carrier of (M | p)
the carrier of (M | p) is set
bool the carrier of (M | p) is non empty set
[: the carrier of (n | (n1 " p)), the carrier of (M | p):] is Relation-like set
bool [: the carrier of (n | (n1 " p)), the carrier of (M | p):] is non empty set
p1 is Relation-like the carrier of (n | (n1 " p)) -defined the carrier of (M | p) -valued Function-like quasi_total Element of bool [: the carrier of (n | (n1 " p)), the carrier of (M | p):]
n is TopSpace-like TopStruct
the carrier of n is set
bool the carrier of n is non empty set
M is TopSpace-like TopStruct
the carrier of M is set
bool the carrier of M is non empty set
p is Element of bool the carrier of n
n1 is Element of bool the carrier of M
n | p is strict TopSpace-like SubSpace of n
M | n1 is strict TopSpace-like SubSpace of M
n is TopSpace-like TopStruct
the carrier of n is set
bool the carrier of n is non empty set
M is TopSpace-like TopStruct
the carrier of M is set
bool the carrier of M is non empty set
p is Element of bool the carrier of n
n1 is Element of bool the carrier of M
n | p is strict TopSpace-like SubSpace of n
M | n1 is strict TopSpace-like SubSpace of M
the carrier of (n | p) is set
the carrier of (M | n1) is set
[: the carrier of (n | p), the carrier of (M | n1):] is Relation-like set
bool [: the carrier of (n | p), the carrier of (M | n1):] is non empty set
p1 is Relation-like the carrier of (n | p) -defined the carrier of (M | n1) -valued Function-like quasi_total Element of bool [: the carrier of (n | p), the carrier of (M | n1):]
dom p1 is Element of bool the carrier of (n | p)
bool the carrier of (n | p) is non empty set
[#] (n | p) is non proper open closed Element of bool the carrier of (n | p)
rng p1 is Element of bool the carrier of (M | n1)
bool the carrier of (M | n1) is non empty set
[#] (M | n1) is non proper open closed Element of bool the carrier of (M | n1)
p1 /" is Relation-like the carrier of (M | n1) -defined the carrier of (n | p) -valued Function-like quasi_total Element of bool [: the carrier of (M | n1), the carrier of (n | p):]
[: the carrier of (M | n1), the carrier of (n | p):] is Relation-like set
bool [: the carrier of (M | n1), the carrier of (n | p):] is non empty set
n is TopSpace-like TopStruct
the carrier of n is set
bool the carrier of n is non empty set
M is TopSpace-like TopStruct
the carrier of M is set
bool the carrier of M is non empty set
p is TopSpace-like TopStruct
the carrier of p is set
bool the carrier of p is non empty set
n1 is Element of bool the carrier of n
p1 is Element of bool the carrier of M
A is Element of bool the carrier of p
n | n1 is strict TopSpace-like SubSpace of n
M | p1 is strict TopSpace-like SubSpace of M
p | A is strict TopSpace-like SubSpace of p
the carrier of (n | n1) is set
the carrier of (p | A) is set
U1 is Relation-like Function-like set
dom U1 is set
rng U1 is set
[: the carrier of (n | n1), the carrier of (p | A):] is Relation-like set
bool [: the carrier of (n | n1), the carrier of (p | A):] is non empty set
U is Relation-like the carrier of (n | n1) -defined the carrier of (p | A) -valued Function-like quasi_total Element of bool [: the carrier of (n | n1), the carrier of (p | A):]
dom U is Element of bool the carrier of (n | n1)
bool the carrier of (n | n1) is non empty set
[#] (n | n1) is non proper open closed Element of bool the carrier of (n | n1)
rng U is Element of bool the carrier of (p | A)
bool the carrier of (p | A) is non empty set
[#] (p | A) is non proper open closed Element of bool the carrier of (p | A)
m is Element of bool the carrier of (p | A)
U " m is Element of bool the carrier of (n | n1)
[:{},{}:] is Relation-like finite set
bool [:{},{}:] is non empty finite V36() set
U /" is Relation-like the carrier of (p | A) -defined the carrier of (n | n1) -valued Function-like quasi_total Element of bool [: the carrier of (p | A), the carrier of (n | n1):]
[: the carrier of (p | A), the carrier of (n | n1):] is Relation-like set
bool [: the carrier of (p | A), the carrier of (n | n1):] is non empty set
S is Element of bool the carrier of (n | n1)
(U /") " S is Element of bool the carrier of (p | A)
n is TopSpace-like TopStruct
M is TopSpace-like TopStruct
weight n is cardinal set
weight M is cardinal set
n is non empty TopSpace-like TopStruct
M is non empty TopSpace-like TopStruct
the carrier of M is non empty set
the carrier of n is non empty set
[: the carrier of M, the carrier of n:] is Relation-like non empty set
bool [: the carrier of M, the carrier of n:] is non empty set
p is Relation-like the carrier of M -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of M, the carrier of n:]
dom p is non empty Element of bool the carrier of M
bool the carrier of M is non empty set
[#] M is non empty non proper open closed Element of bool the carrier of M
rng p is non empty Element of bool the carrier of n
bool the carrier of n is non empty set
[#] n is non empty non proper open closed Element of bool the carrier of n
p /" is Relation-like the carrier of n -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of M:]
[: the carrier of n, the carrier of M:] is Relation-like non empty set
bool [: the carrier of n, the carrier of M:] is non empty set
n1 is Element of the carrier of M
p1 is Element of the carrier of M
p . n1 is Element of the carrier of n
p . p1 is Element of the carrier of n
A is Element of bool the carrier of n
U1 is Element of bool the carrier of n
p " A is Element of bool the carrier of M
p " U1 is Element of bool the carrier of M
U is Element of bool the carrier of M
m is Element of bool the carrier of M
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
M is non empty TopSpace-like TopStruct
p is non empty TopSpace-like TopStruct
the carrier of p is non empty set
the carrier of M is non empty set
[: the carrier of p, the carrier of M:] is Relation-like non empty set
bool [: the carrier of p, the carrier of M:] is non empty set
n1 is Relation-like the carrier of p -defined the carrier of M -valued Function-like non empty total quasi_total Element of bool [: the carrier of p, the carrier of M:]
dom n1 is non empty Element of bool the carrier of p
bool the carrier of p is non empty set
[#] p is non empty non proper open closed Element of bool the carrier of p
rng n1 is non empty Element of bool the carrier of M
bool the carrier of M is non empty set
[#] M is non empty non proper open closed Element of bool the carrier of M
n1 /" is Relation-like the carrier of M -defined the carrier of p -valued Function-like non empty total quasi_total Element of bool [: the carrier of M, the carrier of p:]
[: the carrier of M, the carrier of p:] is Relation-like non empty set
bool [: the carrier of M, the carrier of p:] is non empty set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
p1 is Element of the carrier of p
n1 . p1 is Element of the carrier of M
U1 is a_neighborhood of n1 . p1
U is functional open Element of bool the carrier of (TOP-REAL n)
m is open Element of bool the carrier of M
S is functional open Element of bool the carrier of (TOP-REAL n)
n1 " m is Element of bool the carrier of p
U11 is a_neighborhood of p1
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
M is non empty TopSpace-like TopStruct
p is non empty TopSpace-like TopStruct
n is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
mlt (n,M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
dom (mlt (n,M)) is finite Element of bool NAT
p is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(mlt (n,M)) . p is V21() real ext-real Element of REAL
n /. p is V21() real ext-real Element of REAL
M /. p is V21() real ext-real Element of REAL
(n /. p) * (M /. p) is V21() real ext-real Element of REAL
(mlt (n,M)) /. p is V21() real ext-real Element of REAL
multreal .: (n,M) is set
dom multreal is Relation-like REAL -defined REAL -valued non empty complex-yielding V117() V118() Element of bool [:REAL,REAL:]
rng n is finite V126() V127() V128() Element of bool REAL
rng M is finite V126() V127() V128() Element of bool REAL
[:(rng n),(rng M):] is Relation-like finite complex-yielding V117() V118() set
dom n is finite Element of bool NAT
dom M is finite Element of bool NAT
(dom n) /\ (dom M) is finite Element of bool NAT
M . p is V21() real ext-real Element of REAL
n . p is V21() real ext-real Element of REAL
n is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
len n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
len M is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
p is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
len p is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
len n1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
n ^ p is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
M ^ n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
mlt ((n ^ p),(M ^ n1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
mlt (n,M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
mlt (p,n1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(mlt (n,M)) ^ (mlt (p,n1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
multreal .: ((n ^ p),(M ^ n1)) is set
<:(n ^ p),(M ^ n1):> is Relation-like NAT -defined [:REAL,REAL:] -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of [:REAL,REAL:]
multreal * <:(n ^ p),(M ^ n1):> is Relation-like NAT -defined REAL -valued Function-like finite complex-yielding V117() V118() Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty set
bool [:NAT,REAL:] is non empty set
dom (n ^ p) is finite Element of bool NAT
len (n ^ p) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
Seg (len (n ^ p)) is finite len (n ^ p) -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (n ^ p) ) } is set
dom multreal is Relation-like REAL -defined REAL -valued non empty complex-yielding V117() V118() Element of bool [:REAL,REAL:]
rng (n ^ p) is finite V126() V127() V128() Element of bool REAL
rng (M ^ n1) is finite V126() V127() V128() Element of bool REAL
[:(rng (n ^ p)),(rng (M ^ n1)):] is Relation-like finite complex-yielding V117() V118() set
dom (multreal * <:(n ^ p),(M ^ n1):>) is finite Element of bool NAT
dom (M ^ n1) is finite Element of bool NAT
(dom (n ^ p)) /\ (dom (M ^ n1)) is finite Element of bool NAT
len (M ^ n1) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(len M) + (len n1) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
Seg (len (M ^ n1)) is finite len (M ^ n1) -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (M ^ n1) ) } is set
dom (mlt ((n ^ p),(M ^ n1))) is finite Element of bool NAT
multreal .: (p,n1) is set
<:p,n1:> is Relation-like NAT -defined [:REAL,REAL:] -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of [:REAL,REAL:]
multreal * <:p,n1:> is Relation-like NAT -defined REAL -valued Function-like finite complex-yielding V117() V118() Element of bool [:NAT,REAL:]
rng p is finite V126() V127() V128() Element of bool REAL
rng n1 is finite V126() V127() V128() Element of bool REAL
[:(rng p),(rng n1):] is Relation-like finite complex-yielding V117() V118() set
dom (multreal * <:p,n1:>) is finite Element of bool NAT
dom p is finite Element of bool NAT
dom n1 is finite Element of bool NAT
(dom p) /\ (dom n1) is finite Element of bool NAT
Seg (len p) is finite len p -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len p ) } is set
dom (mlt (p,n1)) is finite Element of bool NAT
len (mlt (p,n1)) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
multreal .: (n,M) is set
<:n,M:> is Relation-like NAT -defined [:REAL,REAL:] -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of [:REAL,REAL:]
multreal * <:n,M:> is Relation-like NAT -defined REAL -valued Function-like finite complex-yielding V117() V118() Element of bool [:NAT,REAL:]
rng n is finite V126() V127() V128() Element of bool REAL
rng M is finite V126() V127() V128() Element of bool REAL
[:(rng n),(rng M):] is Relation-like finite complex-yielding V117() V118() set
dom (multreal * <:n,M:>) is finite Element of bool NAT
dom n is finite Element of bool NAT
dom M is finite Element of bool NAT
(dom n) /\ (dom M) is finite Element of bool NAT
(len n) + (len p) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
Seg (len n) is finite len n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len n ) } is set
dom (mlt (n,M)) is finite Element of bool NAT
len (mlt ((n ^ p),(M ^ n1))) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
p1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
(mlt ((n ^ p),(M ^ n1))) . p1 is V21() real ext-real Element of REAL
((mlt (n,M)) ^ (mlt (p,n1))) . p1 is V21() real ext-real Element of REAL
Seg (len (mlt ((n ^ p),(M ^ n1)))) is finite len (mlt ((n ^ p),(M ^ n1))) -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (mlt ((n ^ p),(M ^ n1))) ) } is set
(n ^ p) /. p1 is V21() real ext-real Element of REAL
(n ^ p) . p1 is V21() real ext-real Element of REAL
(M ^ n1) /. p1 is V21() real ext-real Element of REAL
(M ^ n1) . p1 is V21() real ext-real Element of REAL
M . p1 is V21() real ext-real Element of REAL
(mlt (n,M)) . p1 is V21() real ext-real Element of REAL
n /. p1 is V21() real ext-real Element of REAL
M /. p1 is V21() real ext-real Element of REAL
(n /. p1) * (M /. p1) is V21() real ext-real Element of REAL
n . p1 is V21() real ext-real Element of REAL
((n ^ p) /. p1) * ((M ^ n1) /. p1) is V21() real ext-real Element of REAL
p1 -' (len n) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
p1 - (len n) is V21() real ext-real Element of REAL
(len n) + (p1 -' (len n)) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
((len n) + (len p)) - (len n) is V21() real ext-real Element of REAL
(p1 -' (len n)) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
{} + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
len (mlt (n,M)) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(len (mlt (n,M))) + (p1 -' (len n)) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(mlt (p,n1)) . (p1 -' (len n)) is V21() real ext-real Element of REAL
p /. (p1 -' (len n)) is V21() real ext-real Element of REAL
n1 /. (p1 -' (len n)) is V21() real ext-real Element of REAL
(p /. (p1 -' (len n))) * (n1 /. (p1 -' (len n))) is V21() real ext-real Element of REAL
p . (p1 -' (len n)) is V21() real ext-real Element of REAL
n1 . (p1 -' (len n)) is V21() real ext-real Element of REAL
((n ^ p) /. p1) * ((M ^ n1) /. p1) is V21() real ext-real Element of REAL
((n ^ p) /. p1) * ((M ^ n1) /. p1) is V21() real ext-real Element of REAL
((n ^ p) /. p1) * ((M ^ n1) /. p1) is V21() real ext-real Element of REAL
len (mlt (n,M)) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(len (mlt (n,M))) + (len (mlt (p,n1))) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
len ((mlt (n,M)) ^ (mlt (p,n1))) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
n is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
len n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
len M is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
p is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
len p is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
len n1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
n ^ p is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
M ^ n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|((n ^ p),(M ^ n1))| is V21() real ext-real Element of REAL
|(n,M)| is V21() real ext-real Element of REAL
|(p,n1)| is V21() real ext-real Element of REAL
|(n,M)| + |(p,n1)| is V21() real ext-real Element of REAL
mlt (n,M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
mlt (p,n1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(mlt (n,M)) ^ (mlt (p,n1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum ((mlt (n,M)) ^ (mlt (p,n1))) is V21() real ext-real Element of REAL
Sum (mlt (n,M)) is V21() real ext-real Element of REAL
Sum (mlt (p,n1)) is V21() real ext-real Element of REAL
(Sum (mlt (n,M))) + (Sum (mlt (p,n1))) is V21() real ext-real Element of REAL
mlt ((n ^ p),(M ^ n1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (mlt ((n ^ p),(M ^ n1))) is V21() real ext-real Element of REAL
(Sum (mlt (n,M))) + |(p,n1)| is V21() real ext-real Element of REAL
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
RealVectSpace (Seg n) is non empty V70() V138() V139() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of (RealVectSpace (Seg n)) is non empty set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
M is non empty set
Funcs ((Seg n),M) is functional non empty FUNCTION_DOMAIN of Seg n,M
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
Funcs ((Seg n),REAL) is functional non empty FUNCTION_DOMAIN of Seg n, REAL
RealFuncZero (Seg n) is Relation-like Seg n -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Element of Funcs ((Seg n),REAL)
RealFuncAdd (Seg n) is Relation-like [:(Funcs ((Seg n),REAL)),(Funcs ((Seg n),REAL)):] -defined Funcs ((Seg n),REAL) -valued Function-like non empty total quasi_total Element of bool [:[:(Funcs ((Seg n),REAL)),(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):]
[:(Funcs ((Seg n),REAL)),(Funcs ((Seg n),REAL)):] is Relation-like non empty set
[:[:(Funcs ((Seg n),REAL)),(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):] is Relation-like non empty set
bool [:[:(Funcs ((Seg n),REAL)),(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):] is non empty set
RealFuncExtMult (Seg n) is Relation-like [:REAL,(Funcs ((Seg n),REAL)):] -defined Funcs ((Seg n),REAL) -valued Function-like non empty total quasi_total Element of bool [:[:REAL,(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):]
[:REAL,(Funcs ((Seg n),REAL)):] is Relation-like non empty set
[:[:REAL,(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):] is Relation-like non empty set
bool [:[:REAL,(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):] is non empty set
RLSStruct(# (Funcs ((Seg n),REAL)),(RealFuncZero (Seg n)),(RealFuncAdd (Seg n)),(RealFuncExtMult (Seg n)) #) is strict RLSStruct
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
RealVectSpace (Seg n) is non empty V70() V138() V139() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
0. (RealVectSpace (Seg n)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like V51( RealVectSpace (Seg n)) complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
the carrier of (RealVectSpace (Seg n)) is non empty set
the ZeroF of (RealVectSpace (Seg n)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the carrier of (TOP-REAL n) is functional non empty set
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
Funcs ((Seg n),REAL) is functional non empty FUNCTION_DOMAIN of Seg n, REAL
RealFuncZero (Seg n) is Relation-like Seg n -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Element of Funcs ((Seg n),REAL)
RealFuncAdd (Seg n) is Relation-like [:(Funcs ((Seg n),REAL)),(Funcs ((Seg n),REAL)):] -defined Funcs ((Seg n),REAL) -valued Function-like non empty total quasi_total Element of bool [:[:(Funcs ((Seg n),REAL)),(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):]
[:(Funcs ((Seg n),REAL)),(Funcs ((Seg n),REAL)):] is Relation-like non empty set
[:[:(Funcs ((Seg n),REAL)),(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):] is Relation-like non empty set
bool [:[:(Funcs ((Seg n),REAL)),(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):] is non empty set
RealFuncExtMult (Seg n) is Relation-like [:REAL,(Funcs ((Seg n),REAL)):] -defined Funcs ((Seg n),REAL) -valued Function-like non empty total quasi_total Element of bool [:[:REAL,(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):]
[:REAL,(Funcs ((Seg n),REAL)):] is Relation-like non empty set
[:[:REAL,(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):] is Relation-like non empty set
bool [:[:REAL,(Funcs ((Seg n),REAL)):],(Funcs ((Seg n),REAL)):] is non empty set
RLSStruct(# (Funcs ((Seg n),REAL)),(RealFuncZero (Seg n)),(RealFuncAdd (Seg n)),(RealFuncExtMult (Seg n)) #) is strict RLSStruct
(Seg n) --> {} is Relation-like Seg n -defined {{}} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() V119() Element of bool [:(Seg n),{{}}:]
{{}} is functional non empty finite V36() set
[:(Seg n),{{}}:] is Relation-like finite set
bool [:(Seg n),{{}}:] is non empty finite V36() set
0* n is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of REAL n
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of n -tuples_on REAL
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
M is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL M is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional M -locally_euclidean M -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL M) is functional non empty set
Seg M is finite M -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= M ) } is set
p is Relation-like NAT -defined Function-like finite M -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL M)
p . n is V21() real ext-real Element of REAL
n1 is Relation-like NAT -defined Function-like finite M -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL M)
p + n1 is Relation-like NAT -defined Function-like finite M -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL M)
p + n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(p + n1) . n is V21() real ext-real Element of REAL
n1 . n is V21() real ext-real Element of REAL
(p . n) + (n1 . n) is V21() real ext-real Element of REAL
dom (p + n1) is finite Element of bool NAT
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
RealVectSpace (Seg n) is non empty V70() V138() V139() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of (RealVectSpace (Seg n)) is non empty set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
M is set
bool the carrier of (RealVectSpace (Seg n)) is non empty set
p1 is Relation-like the carrier of (RealVectSpace (Seg n)) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of RealVectSpace (Seg n)
A is finite Element of bool the carrier of (RealVectSpace (Seg n))
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . U1 is V21() real ext-real Element of REAL
Funcs ( the carrier of (TOP-REAL n),REAL) is functional non empty FUNCTION_DOMAIN of the carrier of (TOP-REAL n), REAL
bool the carrier of (TOP-REAL n) is non empty set
p1 is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of TOP-REAL n
A is functional finite closed compact Element of bool the carrier of (TOP-REAL n)
U1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
p1 . U1 is V21() real ext-real Element of REAL
Funcs ( the carrier of (RealVectSpace (Seg n)),REAL) is functional non empty FUNCTION_DOMAIN of the carrier of (RealVectSpace (Seg n)), REAL
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
[: the carrier of (TOP-REAL n),REAL:] is Relation-like non empty set
bool [: the carrier of (TOP-REAL n),REAL:] is non empty set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
RealVectSpace (Seg n) is non empty V70() V138() V139() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of (RealVectSpace (Seg n)) is non empty set
[: the carrier of (RealVectSpace (Seg n)),REAL:] is Relation-like non empty set
bool [: the carrier of (RealVectSpace (Seg n)),REAL:] is non empty set
M 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)
p is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n),REAL:]
p (#) M 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)
n1 is Relation-like NAT -defined the carrier of (RealVectSpace (Seg n)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (RealVectSpace (Seg n))
p1 is Relation-like the carrier of (RealVectSpace (Seg n)) -defined REAL -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (RealVectSpace (Seg n)),REAL:]
p1 (#) n1 is Relation-like NAT -defined the carrier of (RealVectSpace (Seg n)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (RealVectSpace (Seg n))
len (p1 (#) n1) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
len n1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
len (p (#) M) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
len M is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
U1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
M /. U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
n -VectSp_over F_Real is strict VectSpStr over F_Real
the carrier of (n -VectSp_over F_Real) is set
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on the carrier of F_Real is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Real
n1 /. U1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
A is non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of A is non empty set
dom (p1 (#) n1) is finite Element of bool NAT
dom M is finite Element of bool NAT
M . U1 is Relation-like Function-like set
dom n1 is finite Element of bool NAT
n1 . U1 is set
dom (p (#) M) is finite Element of bool NAT
(p (#) M) . U1 is Relation-like Function-like set
S is Element of the carrier of A
p . S is V21() real ext-real Element of REAL
(p . S) * S is Element of the carrier of A
p1 . (n1 /. U1) is V21() real ext-real Element of REAL
(p1 . (n1 /. U1)) * (n1 /. U1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
(p1 . (n1 /. U1)) * (n1 /. U1) is Relation-like Function-like set
(p1 (#) n1) . U1 is set
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
RealVectSpace (Seg n) is non empty V70() V138() V139() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of (RealVectSpace (Seg n)) is non empty set
n1 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)
Sum n1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 is Relation-like NAT -defined the carrier of (RealVectSpace (Seg n)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (RealVectSpace (Seg n))
Sum p1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
A is non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of A is non empty set
[:NAT, the carrier of A:] is Relation-like non empty set
bool [:NAT, the carrier of A:] is non empty set
len n1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
0. A is V51(A) Element of the carrier of A
the ZeroF of A is Element of the carrier of A
U1 is Relation-like NAT -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of A:]
U1 . (len n1) is Element of the carrier of A
U1 . {} is set
[:NAT, the carrier of (RealVectSpace (Seg n)):] is Relation-like non empty set
bool [:NAT, the carrier of (RealVectSpace (Seg n)):] is non empty set
len p1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
0. (RealVectSpace (Seg n)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like V51( RealVectSpace (Seg n)) complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
the ZeroF of (RealVectSpace (Seg n)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
U is Relation-like NAT -defined the carrier of (RealVectSpace (Seg n)) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (RealVectSpace (Seg n)):]
U . (len p1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
U . {} is set
m is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
U1 . m is set
U . m is set
m + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
U1 . (m + 1) is set
U . (m + 1) is set
n -VectSp_over F_Real is strict VectSpStr over F_Real
the carrier of (n -VectSp_over F_Real) is set
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on the carrier of F_Real is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Real
p1 /. (m + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
U . m is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
n1 /. (m + 1) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
dom n1 is finite Element of bool NAT
n1 . (m + 1) is Relation-like Function-like set
U1 . (m + 1) is Element of the carrier of A
U1 . m is Element of the carrier of A
U2 is Element of the carrier of A
(U1 . m) + U2 is Element of the carrier of A
p1 . (m + 1) is set
U11 is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite n -element FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of F_Real
(U . m) + (p1 /. (m + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
K186((U . m),(p1 /. (m + 1))) is Relation-like Function-like set
U . (m + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
RealVectSpace (Seg n) is non empty V70() V138() V139() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of (RealVectSpace (Seg n)) is non empty set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
n1 is Relation-like the carrier of (RealVectSpace (Seg n)) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of RealVectSpace (Seg n)
Sum n1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
p1 is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of TOP-REAL n
Sum p1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
Carrier p1 is functional Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
A 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 A is functional finite closed compact Element of bool the carrier of (TOP-REAL n)
p1 (#) A 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)
Sum (p1 (#) A) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 is Relation-like NAT -defined the carrier of (RealVectSpace (Seg n)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (RealVectSpace (Seg n))
n1 (#) U1 is Relation-like NAT -defined the carrier of (RealVectSpace (Seg n)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (RealVectSpace (Seg n))
Sum (n1 (#) U1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
RealVectSpace (Seg n) is non empty V70() V138() V139() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of (RealVectSpace (Seg n)) is non empty set
bool the carrier of (RealVectSpace (Seg n)) is non empty set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
p is Element of bool the carrier of (RealVectSpace (Seg n))
p1 is functional Element of bool the carrier of (TOP-REAL n)
A is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of p1
Sum A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
0. (RealVectSpace (Seg n)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like V51( RealVectSpace (Seg n)) complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
the ZeroF of (RealVectSpace (Seg n)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
U1 is Relation-like the carrier of (RealVectSpace (Seg n)) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of RealVectSpace (Seg n)
Sum U1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
Carrier A is functional Element of bool the carrier of (TOP-REAL n)
A is Relation-like the carrier of (RealVectSpace (Seg n)) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of p
Carrier A is Element of bool the carrier of (RealVectSpace (Seg n))
U1 is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of TOP-REAL n
Sum A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
U is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of p1
Sum U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
RN_Base n is functional finite FinSequence-membered R-orthogonal R-normal R-orthonormal V353(n) orthogonal_basis Element of bool (REAL n)
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
bool (REAL n) is non empty set
M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p is functional Element of bool the carrier of (TOP-REAL n)
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
RealVectSpace (Seg n) is non empty V70() V138() V139() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of (RealVectSpace (Seg n)) is non empty set
bool the carrier of (RealVectSpace (Seg n)) is non empty set
p1 is Element of bool the carrier of (RealVectSpace (Seg n))
n1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
A is Relation-like the carrier of (RealVectSpace (Seg n)) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of p1
Sum A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (RealVectSpace (Seg n))
Carrier A is Element of bool the carrier of (RealVectSpace (Seg n))
U1 is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of TOP-REAL n
U is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of p
Sum U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
RN_Base n is functional finite FinSequence-membered R-orthogonal R-normal R-orthonormal V353(n) orthogonal_basis Element of bool (REAL n)
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
bool (REAL n) is non empty set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
RealVectSpace (Seg n) is non empty V70() V138() V139() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of (RealVectSpace (Seg n)) is non empty set
bool the carrier of (RealVectSpace (Seg n)) is non empty set
p1 is Element of bool the carrier of (RealVectSpace (Seg n))
n1 is functional finite closed compact Element of bool the carrier of (TOP-REAL n)
(Omega). (TOP-REAL n) is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
Lin n1 is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
A is non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the U5 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 non empty total 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 Relation-like non empty set
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is Relation-like 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 non empty total 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 Relation-like non empty set
[:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is Relation-like non empty set
bool [:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
RLSStruct(# the carrier of (TOP-REAL n), the ZeroF of (TOP-REAL n), the U5 of (TOP-REAL n), the Mult of (TOP-REAL n) #) is strict RLSStruct
U1 is non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of A
the carrier of U1 is non empty set
the carrier of A is non empty set
U is set
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the U5 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 non empty total 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 Relation-like non empty set
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is Relation-like 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 non empty total 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 Relation-like non empty set
[:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is Relation-like non empty set
bool [:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
RLSStruct(# the carrier of (TOP-REAL n), the ZeroF of (TOP-REAL n), the U5 of (TOP-REAL n), the Mult of (TOP-REAL n) #) is strict RLSStruct
m is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
{ (Sum b₁) where b₁ is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of n1 : verum } is set
S is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding V117() V118() Linear_Combination of n1
Sum S is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the U5 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 non empty total 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 Relation-like non empty set
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is Relation-like 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 non empty total 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 Relation-like non empty set
[:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is Relation-like non empty set
bool [:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
RLSStruct(# the carrier of (TOP-REAL n), the ZeroF of (TOP-REAL n), the U5 of (TOP-REAL n), the Mult of (TOP-REAL n) #) is strict RLSStruct
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
the topology of (TOP-REAL n) is non empty open Element of bool (bool the carrier of (TOP-REAL n))
bool (bool the carrier of (TOP-REAL n)) is non empty set
M is functional Element of bool the carrier of (TOP-REAL n)
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) is non empty strict TopSpace-like V234() second-countable TopStruct
Euclid n is non empty strict Reflexive discerning V177() triangle Discerning MetrStruct
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
Pitag_dist n is Relation-like [:(REAL n),(REAL n):] -defined REAL -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [:[:(REAL n),(REAL n):],REAL:]
[:(REAL n),(REAL n):] is Relation-like non empty set
[:[:(REAL n),(REAL n):],REAL:] is Relation-like non empty set
bool [:[:(REAL n),(REAL n):],REAL:] is non empty set
MetrStruct(# (REAL n),(Pitag_dist n) #) is strict MetrStruct
TopSpaceMetr (Euclid n) is metrizable TopStruct
the carrier of (Euclid n) is non empty set
Family_open_set (Euclid n) is Element of bool (bool the carrier of (Euclid n))
bool the carrier of (Euclid n) is non empty set
bool (bool the carrier of (Euclid n)) is non empty set
TopStruct(# the carrier of (Euclid n),(Family_open_set (Euclid n)) #) is non empty strict TopStruct
A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 is Element of the carrier of (Euclid n)
p1 is Element of bool the carrier of (Euclid n)
U is V21() real ext-real Element of REAL
Ball (U1,U) is Element of bool the carrier of (Euclid n)
m is V21() real ext-real set
S is V21() real ext-real set
p is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
Euclid p is non empty strict Reflexive discerning V177() triangle Discerning MetrStruct
REAL p is functional non empty FinSequence-membered FinSequenceSet of REAL
p -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
Pitag_dist p is Relation-like [:(REAL p),(REAL p):] -defined REAL -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [:[:(REAL p),(REAL p):],REAL:]
[:(REAL p),(REAL p):] is Relation-like non empty set
[:[:(REAL p),(REAL p):],REAL:] is Relation-like non empty set
bool [:[:(REAL p),(REAL p):],REAL:] is non empty set
MetrStruct(# (REAL p),(Pitag_dist p) #) is strict MetrStruct
the carrier of (Euclid p) is non empty set
TOP-REAL p is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional p -locally_euclidean p -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL p) is functional non empty set
U11 is Element of the carrier of (Euclid p)
Ball (U11,S) is Element of bool the carrier of (Euclid p)
bool the carrier of (Euclid p) is non empty set
V is Relation-like NAT -defined Function-like finite p -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL p)
Ball (V,S) is functional open V199( TOP-REAL p) V362( TOP-REAL p) Element of bool the carrier of (TOP-REAL p)
bool the carrier of (TOP-REAL p) is non empty set
Ball (A,S) is functional open V362( TOP-REAL n) Element of bool the carrier of (TOP-REAL n)
A is Element of the carrier of (Euclid n)
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U is V21() real ext-real set
Ball (U1,U) is functional open V362( TOP-REAL n) Element of bool the carrier of (TOP-REAL n)
m is V21() real ext-real Element of REAL
Ball (A,m) is Element of bool the carrier of (Euclid n)
S is Element of the carrier of (Euclid p)
Ball (S,m) is Element of bool the carrier of (Euclid p)
U11 is Relation-like NAT -defined Function-like finite p -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL p)
Ball (U11,m) is functional open V199( TOP-REAL p) V362( TOP-REAL p) Element of bool the carrier of (TOP-REAL p)
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
[: the carrier of (TOP-REAL n), the carrier of R^1:] is Relation-like non empty complex-yielding V117() V118() set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p is set
n1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(M,n1)| is V21() real ext-real Element of REAL
p is Relation-like Function-like set
dom p is set
rng p is set
n1 is set
p1 is set
p . p1 is set
A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(M,A)| is V21() real ext-real Element of REAL
n1 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
p1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p . p1 is set
|(M,p1)| is V21() real ext-real Element of REAL
A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(M,A)| is V21() real ext-real Element of REAL
p1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
n1 . p1 is V21() real ext-real Element of REAL
|(M,p1)| is V21() real ext-real Element of REAL
p is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
n1 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
p1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p . p1 is V21() real ext-real Element of REAL
n1 . p1 is V21() real ext-real Element of REAL
|(M,p1)| is V21() real ext-real Element of REAL
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(n,M) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() 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 Relation-like non empty complex-yielding V117() V118() set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
the topology of (TOP-REAL n) is non empty open 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 V234() second-countable TopStruct
Euclid n is non empty strict Reflexive discerning V177() triangle Discerning MetrStruct
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
Pitag_dist n is Relation-like [:(REAL n),(REAL n):] -defined REAL -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [:[:(REAL n),(REAL n):],REAL:]
[:(REAL n),(REAL n):] is Relation-like non empty set
[:[:(REAL n),(REAL n):],REAL:] is Relation-like non empty set
bool [:[:(REAL n),(REAL n):],REAL:] is non empty set
MetrStruct(# (REAL n),(Pitag_dist n) #) is strict MetrStruct
TopSpaceMetr (Euclid n) is metrizable TopStruct
the carrier of (TopSpaceMetr (Euclid n)) is set
the carrier of (TopSpaceMetr RealSpace) is set
[: the carrier of (TopSpaceMetr (Euclid n)), the carrier of (TopSpaceMetr RealSpace):] is Relation-like set
bool [: the carrier of (TopSpaceMetr (Euclid n)), the carrier of (TopSpaceMetr RealSpace):] is non empty set
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(n,M) . p1 is V21() real ext-real Element of REAL
|(p1,M)| is V21() real ext-real Element of REAL
p1 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
A is set
(n,M) . A is V21() real ext-real Element of REAL
p1 . A is V21() real ext-real Element of REAL
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(n,M) . U1 is V21() real ext-real Element of REAL
p1 . U1 is V21() real ext-real Element of REAL
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|.M.| is V21() real ext-real non negative Element of REAL
sqr M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr M) is V21() real ext-real Element of REAL
sqrt (Sum (sqr M)) is V21() real ext-real Element of REAL
the carrier of (Euclid n) is non empty set
the carrier of RealSpace is non empty V128() set
A is V21() real ext-real set
U is Element of the carrier of RealSpace
n1 is Relation-like the carrier of (TopSpaceMetr (Euclid n)) -defined the carrier of (TopSpaceMetr RealSpace) -valued Function-like quasi_total Element of bool [: the carrier of (TopSpaceMetr (Euclid n)), the carrier of (TopSpaceMetr RealSpace):]
U1 is Element of the carrier of (Euclid n)
n1 . U1 is set
A / |.M.| is V21() real ext-real Element of REAL
S is Element of the carrier of (Euclid n)
n1 . S is set
dist (U1,S) is V21() real ext-real Element of REAL
U11 is Element of the carrier of RealSpace
dist (U,U11) is V21() real ext-real Element of REAL
p1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
TOP-REAL p1 is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional p1 -locally_euclidean p1 -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL p1) is functional non empty set
the distance of RealSpace is Relation-like [: the carrier of RealSpace, the carrier of RealSpace:] -defined REAL -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [:[: the carrier of RealSpace, the carrier of RealSpace:],REAL:]
[: the carrier of RealSpace, the carrier of RealSpace:] is Relation-like non empty complex-yielding V117() V118() set
[:[: the carrier of RealSpace, the carrier of RealSpace:],REAL:] is Relation-like non empty set
bool [:[: the carrier of RealSpace, the carrier of RealSpace:],REAL:] is non empty set
the distance of RealSpace . (U,U11) is V21() real ext-real Element of REAL
V is V21() real ext-real Element of REAL
U2 is V21() real ext-real Element of REAL
V - U2 is V21() real ext-real Element of REAL
abs (V - U2) is V21() real ext-real Element of REAL
p2 is Relation-like NAT -defined Function-like finite p1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL p1)
|(p2,M)| is V21() real ext-real Element of REAL
g4 is Relation-like NAT -defined Function-like finite p1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL p1)
|(g4,M)| is V21() real ext-real Element of REAL
|(g4,M)| - |(p2,M)| is V21() real ext-real Element of REAL
abs (|(g4,M)| - |(p2,M)|) is V21() real ext-real Element of REAL
g4 - p2 is Relation-like NAT -defined Function-like finite p1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL p1)
g4 - p2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|((g4 - p2),M)| is V21() real ext-real Element of REAL
abs |((g4 - p2),M)| is V21() real ext-real Element of REAL
|.(g4 - p2).| is V21() real ext-real non negative Element of REAL
sqr (g4 - p2) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g4 - p2)) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g4 - p2))) is V21() real ext-real Element of REAL
|.(g4 - p2).| * |.M.| is V21() real ext-real non negative Element of REAL
(dist (U1,S)) * |.M.| is V21() real ext-real Element of REAL
(A / |.M.|) * |.M.| is V21() real ext-real Element of REAL
|.M.| / |.M.| is V21() real ext-real non negative Element of REAL
A / (|.M.| / |.M.|) is V21() real ext-real Element of REAL
A / 1 is V21() real ext-real Element of REAL
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : |(M,(b₁ - p))| = {} } is set
bool the carrier of (TOP-REAL n) is non empty set
p1 is set
A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
A - p is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
A - p is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(A - p))| is V21() real ext-real Element of REAL
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
transl (M,(TOP-REAL n)) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like non empty total 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 Relation-like non empty set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
p is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
n1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(n,p,n1) is functional Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : |(p,(b₁ - n1))| = {} } is set
(transl (M,(TOP-REAL n))) .: (n,p,n1) is functional Element of bool the carrier of (TOP-REAL n)
M + n1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
M + n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(n,p,(M + n1)) is functional Element of bool the carrier of (TOP-REAL n)
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : |(p,(b₁ - (M + n1)))| = {} } is set
p1 is set
A is set
[A,p1] is set
{A,p1} is non empty finite set
{A} is non empty finite set
{{A,p1},{A}} is non empty finite V36() set
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 - n1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 - n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(p,(U1 - n1))| is V21() real ext-real Element of REAL
(transl (M,(TOP-REAL n))) . U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
M + U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
M + U1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
dom (transl (M,(TOP-REAL n))) is functional non empty Element of bool the carrier of (TOP-REAL n)
(transl (M,(TOP-REAL n))) . A is Relation-like Function-like set
rng (transl (M,(TOP-REAL n))) is functional non empty Element of bool the carrier of (TOP-REAL n)
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(U1 - n1) + (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(U1 - n1) + (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
- M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
- M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
M + (- M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
M + (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(U1 - n1) + (M + (- M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(U1 - n1) + (M + (- M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
- n1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
- n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U1 + (- n1) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 + (- n1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(U1 + (- n1)) + (M + (- M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(U1 + (- n1)) + (M + (- M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(U1 + (- n1)) + M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(U1 + (- n1)) + M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((U1 + (- n1)) + M) + (- M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((U1 + (- n1)) + M) + (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U + (- n1) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U + (- n1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(U + (- n1)) + (- M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(U + (- n1)) + (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U - n1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U - n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(U - n1) + (- M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(U - n1) + (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(U - n1) - M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(U - n1) - M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U - (M + n1) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U - (M + n1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
p1 is set
A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
A - (M + n1) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
A - (M + n1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(p,(A - (M + n1)))| is V21() real ext-real Element of REAL
A - M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
A - M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
M + (A - M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
M + (A - M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(transl (M,(TOP-REAL n))) . (A - M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
[(A - M),A] is set
{(A - M),A} is functional non empty finite V36() set
{(A - M)} is functional non empty finite V36() set
{{(A - M),A},{(A - M)}} is non empty finite V36() set
(A - M) - n1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(A - M) - n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
n - 1 is V21() real ext-real Element of REAL
M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(n,M,(0. (TOP-REAL n))) is functional Element of bool the carrier of (TOP-REAL n)
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : |(M,(b₁ - (0. (TOP-REAL n))))| = {} } is set
p is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
TOP-REAL p is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional p -locally_euclidean p -manifold manifold-like RLTopStruct
0. (TOP-REAL p) is Relation-like NAT -defined Function-like finite p -element FinSequence-like FinSubsequence-like V51( TOP-REAL p) complex-yielding V117() V118() Element of the carrier of (TOP-REAL p)
the carrier of (TOP-REAL p) is functional non empty set
the ZeroF of (TOP-REAL p) is Relation-like NAT -defined Function-like finite p -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL p)
- (0. (TOP-REAL p)) is Relation-like NAT -defined Function-like finite p -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL p)
- (0. (TOP-REAL p)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
- (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
- (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(0. (TOP-REAL n)))| is V21() real ext-real Element of REAL
(0. (TOP-REAL n)) - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(0. (TOP-REAL n)) - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,((0. (TOP-REAL n)) - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
p1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 + A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 + A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(U1 - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(U - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
U1 + U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 + U is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(U1 + U) - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(U1 + U) - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,((U1 + U) - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
U1 + (U - (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 + (U - (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(U1 + (U - (0. (TOP-REAL n)))))| is V21() real ext-real Element of REAL
|(M,U1)| is V21() real ext-real Element of REAL
|(M,U1)| + |(M,(U - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
U1 + (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 + (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(U1 + (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
p1 is V21() real ext-real Element of REAL
A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 * A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(U1 - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
p1 * (U1 - (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 * (U1 - (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(p1 * (U1 - (0. (TOP-REAL n)))))| is V21() real ext-real Element of REAL
p1 * {} is V21() real ext-real Element of REAL
p1 * U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 * U1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
p1 * (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 * (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(p1 * U1) - (p1 * (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(p1 * U1) - (p1 * (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,((p1 * U1) - (p1 * (0. (TOP-REAL n)))))| is V21() real ext-real Element of REAL
(p1 * U1) - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(p1 * U1) - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,((p1 * U1) - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
p1 is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
the carrier of p1 is non empty set
the Basis of p1 is Basis of p1
Lin the Basis of p1 is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of p1
the carrier of (Lin the Basis of p1) is non empty set
[#] (Lin the Basis of p1) is non empty non proper Element of bool the carrier of (Lin the Basis of p1)
bool the carrier of (Lin the Basis of p1) is non empty set
U1 is functional linearly-independent Element of bool the carrier of (TOP-REAL n)
card U1 is cardinal set
Lin U1 is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
[#] (Lin U1) is non empty non proper Element of bool the carrier of (Lin U1)
the carrier of (Lin U1) is non empty set
bool the carrier of (Lin U1) is non empty set
{M} is functional non empty finite V36() set
U is functional linearly-independent Element of bool the carrier of (TOP-REAL n)
Lin U is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
dim (Lin U) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
card U is cardinal set
m is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(M,m)| is V21() real ext-real Element of REAL
|(M,M)| is V21() real ext-real Element of REAL
|(M,m)| / |(M,M)| is V21() real ext-real Element of REAL
(|(M,m)| / |(M,M)|) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(|(M,m)| / |(M,M)|) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
m - ((|(M,m)| / |(M,M)|) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
m - ((|(M,m)| / |(M,M)|) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 + p2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 + p2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,U2)| is V21() real ext-real Element of REAL
|(M,p2)| is V21() real ext-real Element of REAL
|(M,m)| - |(M,p2)| is V21() real ext-real Element of REAL
(|(M,m)| / |(M,M)|) * |(M,M)| is V21() real ext-real Element of REAL
|(M,m)| - ((|(M,m)| / |(M,M)|) * |(M,M)|) is V21() real ext-real Element of REAL
|(M,m)| - |(M,m)| is V21() real ext-real Element of REAL
U2 + (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 + (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(U2 + (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
U2 - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(U2 - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
the U5 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 non empty total 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 Relation-like non empty set
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is Relation-like 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 non empty total 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 Relation-like non empty set
[:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is Relation-like non empty set
bool [:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
RLSStruct(# the carrier of (TOP-REAL n), the ZeroF of (TOP-REAL n), the U5 of (TOP-REAL n), the Mult of (TOP-REAL n) #) is strict RLSStruct
(Lin U1) + (Lin U) is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
(Lin U1) /\ (Lin U) is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
(0). (TOP-REAL n) is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
m is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
S is V21() real ext-real Element of REAL
S * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
S * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U11 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U11 - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U11 - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(U11 - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
m + (- (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
m + (- (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(m + (- (0. (TOP-REAL n)))))| is V21() real ext-real Element of REAL
|(M,m)| is V21() real ext-real Element of REAL
|(M,M)| is V21() real ext-real Element of REAL
S * |(M,M)| is V21() real ext-real Element of REAL
dim (TOP-REAL n) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
dim (Lin U1) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(dim (Lin U1)) + (dim (Lin U)) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(dim (Lin U1)) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is Relation-like non empty set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(n,M,(0. (TOP-REAL n))) is functional Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : |(M,(b₁ - (0. (TOP-REAL n))))| = {} } is set
p is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(n,p,(0. (TOP-REAL n))) is functional Element of bool the carrier of (TOP-REAL n)
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : |(p,(b₁ - (0. (TOP-REAL n))))| = {} } is set
n - 1 is V21() real ext-real Element of REAL
n1 is functional linearly-independent Element of bool the carrier of (TOP-REAL n)
card n1 is cardinal set
Lin n1 is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
[#] (Lin n1) is non empty non proper Element of bool the carrier of (Lin n1)
the carrier of (Lin n1) is non empty set
bool the carrier of (Lin n1) is non empty set
p1 is functional linearly-independent Element of bool the carrier of (TOP-REAL n)
card p1 is cardinal set
Lin p1 is non empty V70() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
[#] (Lin p1) is non empty non proper Element of bool the carrier of (Lin p1)
the carrier of (Lin p1) is non empty set
bool the carrier of (Lin p1) is non empty set
A 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 A is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran A) .: ([#] (Lin n1)) is functional Element of bool the carrier of (TOP-REAL n)
U1 is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like tabular invertible Matrix of n,n, the carrier of F_Real
Mx2Tran U1 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like non empty total quasi_total being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran U1) .: (n,M,(0. (TOP-REAL n))) is functional Element of bool the carrier of (TOP-REAL n)
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(n,M,p) is functional Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : |(M,(b₁ - p))| = {} } is set
(TOP-REAL n) | (n,M,p) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL n
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(M,(0. (TOP-REAL n)))| is V21() real ext-real Element of REAL
p - p is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p - p is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(p - p))| is V21() real ext-real Element of REAL
<*1*> is Relation-like NAT -defined NAT -valued Function-like non empty finite 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() V119() FinSequence of NAT
[1,1] is set
{1,1} is non empty finite set
{{1,1},{1}} is non empty finite V36() set
{[1,1]} is Relation-like Function-like one-to-one non empty finite set
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
n + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
Base_FinSeq ((n + 1),(n + 1)) is Relation-like NAT -defined REAL -valued Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of REAL (n + 1)
REAL (n + 1) is functional non empty FinSequence-membered FinSequenceSet of REAL
(n + 1) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the carrier of (TOP-REAL n) is functional non empty set
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(0. (TOP-REAL n)) ^ <*1*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
K174(n,1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
dom (Base_FinSeq ((n + 1),(n + 1))) is finite Element of bool NAT
Seg (n + 1) is non empty finite n + 1 -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + 1 ) } is set
len <*1*> is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
n + (len <*1*>) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
Seg (n + (len <*1*>)) is finite n + (len <*1*>) -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + (len <*1*>) ) } is set
len (0. (TOP-REAL n)) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(len (0. (TOP-REAL n))) + (len <*1*>) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
Seg ((len (0. (TOP-REAL n))) + (len <*1*>)) is finite (len (0. (TOP-REAL n))) + (len <*1*>) -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= (len (0. (TOP-REAL n))) + (len <*1*>) ) } is set
dom (0. (TOP-REAL n)) is finite Element of bool NAT
p1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
(Base_FinSeq ((n + 1),(n + 1))) . p1 is V21() real ext-real Element of REAL
(0. (TOP-REAL n)) . p1 is V21() real ext-real Element of REAL
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
{} + n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
1 + n is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
0* n is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of REAL n
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of n -tuples_on REAL
(0* n) . p1 is V21() real ext-real Element of REAL
dom <*1*> is non empty finite Element of bool NAT
p1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
(len (0. (TOP-REAL n))) + p1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(Base_FinSeq ((n + 1),(n + 1))) . ((len (0. (TOP-REAL n))) + p1) is V21() real ext-real Element of REAL
<*1*> . p1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of REAL
{} + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
(Base_FinSeq ((n + 1),(n + 1))) . (n + 1) is V21() real ext-real Element of REAL
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
n + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
TOP-REAL (n + 1) is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n + 1 -locally_euclidean n + 1 -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL (n + 1)) is functional non empty set
Base_FinSeq ((n + 1),(n + 1)) is Relation-like NAT -defined REAL -valued Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of REAL (n + 1)
REAL (n + 1) is functional non empty FinSequence-membered FinSequenceSet of REAL
(n + 1) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
0. (TOP-REAL (n + 1)) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like V51( TOP-REAL (n + 1)) complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
the ZeroF of (TOP-REAL (n + 1)) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
M is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
((n + 1),M,(0. (TOP-REAL (n + 1)))) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL (n + 1)
((n + 1),M,(0. (TOP-REAL (n + 1)))) is functional Element of bool the carrier of (TOP-REAL (n + 1))
bool the carrier of (TOP-REAL (n + 1)) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1)) : |(M,(b₁ - (0. (TOP-REAL (n + 1)))))| = {} } is set
(TOP-REAL (n + 1)) | ((n + 1),M,(0. (TOP-REAL (n + 1)))) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL (n + 1)
the carrier of (TOP-REAL n) is functional non empty set
<*{}*> is Relation-like NAT -defined Function-like non empty finite 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
[1,{}] is set
{1,{}} is non empty finite set
{{1,{}},{1}} is non empty finite V36() set
{[1,{}]} is Relation-like Function-like one-to-one non empty finite set
the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1)))) is non empty set
p1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
K174(n,1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
len p1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
len (p1 ^ <*{}*>) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
rng (p1 ^ <*{}*>) is non empty finite V126() V127() V128() Element of bool REAL
dom M is finite Element of bool NAT
Seg (n + 1) is non empty finite n + 1 -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + 1 ) } is set
A is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
dom A is finite Element of bool NAT
(dom M) /\ (dom A) is finite Element of bool NAT
mlt (M,A) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
dom (mlt (M,A)) is finite Element of bool NAT
rng (mlt (M,A)) is finite V126() V127() V128() Element of bool REAL
U1 is V21() real ext-real Element of REAL
U is set
S is set
U11 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(mlt (M,A)) . S is V21() real ext-real Element of REAL
M . S is V21() real ext-real Element of REAL
A . S is V21() real ext-real Element of REAL
(M . S) * (A . S) is V21() real ext-real Element of REAL
<*U1*> is Relation-like NAT -defined REAL -valued Function-like non empty finite 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
[1,U1] is set
{1,U1} is non empty finite set
{{1,U1},{1}} is non empty finite V36() set
{[1,U1]} is Relation-like Function-like one-to-one non empty finite set
len <*U1*> is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
(len p1) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
A . ((len p1) + 1) is V21() real ext-real Element of REAL
<*U1*> . 1 is V21() real ext-real Element of REAL
S is set
(mlt (M,A)) . S is V21() real ext-real Element of REAL
S is set
(mlt (M,A)) . S is V21() real ext-real Element of REAL
{U1} is non empty finite set
(Seg (n + 1)) --> U1 is Relation-like Seg (n + 1) -defined REAL -valued Function-like non empty total quasi_total finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of bool [:(Seg (n + 1)),REAL:]
[:(Seg (n + 1)),REAL:] is Relation-like non empty set
bool [:(Seg (n + 1)),REAL:] is non empty set
(n + 1) |-> U1 is Relation-like NAT -defined REAL -valued Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of (n + 1) -tuples_on REAL
|(M,A)| is V21() real ext-real Element of REAL
Sum (mlt (M,A)) is V21() real ext-real Element of REAL
(n + 1) * U1 is V21() real ext-real Element of REAL
A - (0. (TOP-REAL (n + 1))) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
A - (0. (TOP-REAL (n + 1))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(A - (0. (TOP-REAL (n + 1)))))| is V21() real ext-real Element of REAL
|(M,(0. (TOP-REAL (n + 1))))| is V21() real ext-real Element of REAL
|(M,A)| - |(M,(0. (TOP-REAL (n + 1))))| is V21() real ext-real Element of REAL
|(M,A)| - {} is V21() real ext-real Element of REAL
[#] ((n + 1),M,(0. (TOP-REAL (n + 1)))) is non empty non proper open closed Element of bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1)))) is non empty set
p1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
K174(n,1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
U1 is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
[: the carrier of (TOP-REAL n), the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1)))):] is Relation-like non empty set
bool [: the carrier of (TOP-REAL n), the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1)))):] is non empty set
p1 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1)))) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (TOP-REAL n), the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1)))):]
dom p1 is functional non empty Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
[#] ((n + 1),M,(0. (TOP-REAL (n + 1)))) is non empty non proper open closed Element of bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1)))) is non empty set
A is set
U1 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
U1 - (0. (TOP-REAL (n + 1))) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
U1 - (0. (TOP-REAL (n + 1))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(U1 - (0. (TOP-REAL (n + 1)))))| is V21() real ext-real Element of REAL
len U1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
rng U1 is finite V126() V127() V128() Element of bool REAL
U is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U | n is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
U | (Seg n) is Relation-like NAT -defined Seg n -defined NAT -defined REAL -valued Function-like finite FinSubsequence-like complex-yielding V117() V118() set
S is set
{} + n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
1 + n is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
len (U | n) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
rng (U | n) is finite V126() V127() V128() Element of bool 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
U11 is set
V is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . V is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
U2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
K174(n,1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
p1 . U11 is set
(U | n) ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
U1 . (n + 1) is V21() real ext-real Element of REAL
<*(U1 . (n + 1))*> is Relation-like NAT -defined REAL -valued Function-like non empty finite 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
[1,(U1 . (n + 1))] is set
{1,(U1 . (n + 1))} is non empty finite set
{{1,(U1 . (n + 1))},{1}} is non empty finite V36() set
{[1,(U1 . (n + 1))]} is Relation-like Function-like one-to-one non empty finite set
(U | n) ^ <*(U1 . (n + 1))*> is Relation-like NAT -defined REAL -valued Function-like non empty finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,U1)| is V21() real ext-real Element of REAL
|(M,(0. (TOP-REAL (n + 1))))| is V21() real ext-real Element of REAL
|(M,U1)| - |(M,(0. (TOP-REAL (n + 1))))| is V21() real ext-real Element of REAL
|(M,U1)| - {} is V21() real ext-real Element of REAL
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
rng <*1*> is non empty finite V126() V127() V128() V131() Element of bool REAL
U2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
p2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U2 ^ p2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
g4 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(U | n) ^ g4 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|((U2 ^ p2),((U | n) ^ g4))| is V21() real ext-real Element of REAL
g6 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
Seg g6 is finite g6 -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= g6 ) } is set
Funcs ((Seg g6),REAL) is functional non empty FUNCTION_DOMAIN of Seg g6, REAL
g7 is Relation-like Function-like set
dom g7 is set
rng g7 is set
len U2 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
len p2 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
len g4 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
g5 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|((0. (TOP-REAL n)),g5)| is V21() real ext-real Element of REAL
|(p2,g4)| is V21() real ext-real Element of REAL
|((0. (TOP-REAL n)),g5)| + |(p2,g4)| is V21() real ext-real Element of REAL
{} + |(p2,g4)| is V21() real ext-real Element of REAL
mlt (p2,g4) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (mlt (p2,g4)) is V21() real ext-real Element of REAL
1 * (U1 . (n + 1)) is V21() real ext-real Element of REAL
<*(1 * (U1 . (n + 1)))*> is Relation-like NAT -defined REAL -valued Function-like non empty finite 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
[1,(1 * (U1 . (n + 1)))] is set
{1,(1 * (U1 . (n + 1)))} is non empty finite set
{{1,(1 * (U1 . (n + 1)))},{1}} is non empty finite V36() set
{[1,(1 * (U1 . (n + 1)))]} is Relation-like Function-like one-to-one non empty finite set
Sum <*(1 * (U1 . (n + 1)))*> is V21() real ext-real Element of REAL
U1 is set
p1 . U1 is set
U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . U is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
m is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
m ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
K174(n,1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
rng p1 is non empty Element of bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
A is set
U1 is set
p1 . A is set
p1 . U1 is set
U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . U is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
U ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
K174(n,1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
m is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . m is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
m ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . U1 is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
[#] (TOP-REAL (n + 1)) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL (n + 1))
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . U1 is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
|.U1.| is V21() real ext-real non negative Element of REAL
sqr U1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr U1) is V21() real ext-real Element of REAL
sqrt (Sum (sqr U1)) is V21() real ext-real Element of REAL
U is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
|.U.| is V21() real ext-real non negative Element of REAL
sqr U is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr U) is V21() real ext-real Element of REAL
sqrt (Sum (sqr U)) is V21() real ext-real Element of REAL
U1 ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
K174(n,1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
len U1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
rng U1 is finite V126() V127() V128() Element of bool REAL
m is Relation-like NAT -defined REAL -valued Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of REAL (n + 1)
m | n is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
m | (Seg n) is Relation-like NAT -defined Seg n -defined NAT -defined REAL -valued Function-like finite FinSubsequence-like complex-yielding V117() V118() set
dom U1 is finite Element of bool NAT
(U1 ^ <*{}*>) | (dom U1) is Relation-like NAT -defined dom U1 -defined NAT -defined Function-like finite FinSubsequence-like complex-yielding V117() V118() set
m . (n + 1) is V21() real ext-real Element of REAL
|.U.| ^2 is V21() real ext-real Element of REAL
|.U1.| ^2 is V21() real ext-real Element of REAL
{} ^2 is V21() real ext-real set
(|.U1.| ^2) + ({} ^2) is V21() real ext-real Element of REAL
{} * {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural real finite finite-yielding V36() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding V117() V118() V119() R-orthogonal R-normal R-orthonormal Element of NAT
(|.U1.| ^2) + ({} * {}) is V21() real ext-real Element of REAL
|.U.| * |.U.| is V21() real ext-real non negative Element of REAL
|.U1.| * |.U1.| is V21() real ext-real non negative Element of REAL
sqrt (|.U.| ^2) is V21() real ext-real Element of REAL
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . U1 is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . U is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
U1 - U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 - U is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
p1 . (U1 - U) is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
m is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
S is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
m - S is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
m - S is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U1 ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
K174(n,1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
U ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
(U1 - U) ^ <*{}*> is Relation-like NAT -defined Function-like non empty finite K174(n,1) -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() set
dom ((U1 - U) ^ <*{}*>) is non empty finite Element of bool NAT
Seg (n + 1) is non empty finite n + 1 -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + 1 ) } is set
dom (m - S) is finite Element of bool NAT
V is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
(m - S) . V is V21() real ext-real Element of REAL
((U1 - U) ^ <*{}*>) . V is V21() real ext-real Element of REAL
- S is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
- S is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
m + (- S) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
m + (- S) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(m + (- S)) . V is V21() real ext-real Element of REAL
m . V is V21() real ext-real Element of REAL
(- S) . V is V21() real ext-real Element of REAL
(m . V) + ((- S) . V) is V21() real ext-real Element of REAL
- 1 is non empty V21() real ext-real non positive negative Element of REAL
(- 1) * S is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
(- 1) * S is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((- 1) * S) . V is V21() real ext-real Element of REAL
(m . V) + (((- 1) * S) . V) is V21() real ext-real Element of REAL
S . V is V21() real ext-real Element of REAL
(- 1) * (S . V) is V21() real ext-real Element of REAL
(m . V) + ((- 1) * (S . V)) is V21() real ext-real Element of REAL
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
{(n + 1)} is non empty finite set
(Seg n) \/ {(n + 1)} is non empty finite set
dom U1 is finite Element of bool NAT
dom U is finite Element of bool NAT
U1 . V is V21() real ext-real Element of REAL
U . V is V21() real ext-real Element of REAL
dom (U1 - U) is finite Element of bool NAT
(U1 - U) . V is V21() real ext-real Element of REAL
- U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
- U is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U1 + (- U) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 + (- U) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(U1 + (- U)) . V is V21() real ext-real Element of REAL
(- U) . V is V21() real ext-real Element of REAL
(U1 . V) + ((- U) . V) is V21() real ext-real Element of REAL
(- 1) * U is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(- 1) * U is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((- 1) * U) . V is V21() real ext-real Element of REAL
(U1 . V) + (((- 1) * U) . V) is V21() real ext-real Element of REAL
(- 1) * (U . V) is V21() real ext-real Element of REAL
(U1 . V) + ((- 1) * (U . V)) is V21() real ext-real Element of REAL
len U1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
len U is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
len (U1 - U) is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
p1 .: ([#] (TOP-REAL n)) is Element of bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
U1 is functional Element of bool the carrier of (TOP-REAL n)
p1 .: U1 is Element of bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
([#] (TOP-REAL n)) \ U1 is functional Element of bool the carrier of (TOP-REAL n)
the topology of (TOP-REAL n) is non empty open Element of bool (bool the carrier of (TOP-REAL n))
bool (bool the carrier of (TOP-REAL n)) is non empty set
{ (Ball (b₁,b₂)) where b₁ is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1)), b₂ is V21() real ext-real Element of REAL : ex b₃ being Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) st
( b₁ = p1 . b₃ & Ball (b₃,b₂) c= ([#] (TOP-REAL n)) \ U1 ) } is set
[#] (TOP-REAL (n + 1)) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL (n + 1))
bool ([#] (TOP-REAL (n + 1))) is non empty set
m is set
S is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
U11 is V21() real ext-real Element of REAL
Ball (S,U11) is functional open V199( TOP-REAL (n + 1)) V362( TOP-REAL (n + 1)) Element of bool the carrier of (TOP-REAL (n + 1))
V is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . V is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
Ball (V,U11) is functional open V362( TOP-REAL n) Element of bool the carrier of (TOP-REAL n)
bool (bool the carrier of (TOP-REAL (n + 1))) is non empty set
m is Element of bool (bool the carrier of (TOP-REAL (n + 1)))
S is functional Element of bool the carrier of (TOP-REAL (n + 1))
U11 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
V is V21() real ext-real Element of REAL
Ball (U11,V) is functional open V199( TOP-REAL (n + 1)) V362( TOP-REAL (n + 1)) Element of bool the carrier of (TOP-REAL (n + 1))
U2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . U2 is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
Ball (U2,V) is functional open V362( TOP-REAL n) Element of bool the carrier of (TOP-REAL n)
union m is functional Element of bool the carrier of (TOP-REAL (n + 1))
the topology of (TOP-REAL (n + 1)) is non empty open Element of bool (bool the carrier of (TOP-REAL (n + 1)))
p1 .: (([#] (TOP-REAL n)) \ U1) is Element of bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
(union m) /\ ([#] ((n + 1),M,(0. (TOP-REAL (n + 1))))) is Element of bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
U11 is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
V is set
p1 . V is set
U2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
g4 is V21() real ext-real set
Ball (U2,g4) is functional open V362( TOP-REAL n) Element of bool the carrier of (TOP-REAL n)
p2 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
g5 is V21() real ext-real Element of REAL
Ball (p2,g5) is functional open V199( TOP-REAL (n + 1)) V362( TOP-REAL (n + 1)) Element of bool the carrier of (TOP-REAL (n + 1))
V is set
U2 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
p2 is V21() real ext-real Element of REAL
Ball (U2,p2) is functional open V199( TOP-REAL (n + 1)) V362( TOP-REAL (n + 1)) Element of bool the carrier of (TOP-REAL (n + 1))
g4 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . g4 is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
Ball (g4,p2) is functional open V362( TOP-REAL n) Element of bool the carrier of (TOP-REAL n)
g4 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . g4 is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
Ball (g4,p2) is functional open V362( TOP-REAL n) Element of bool the carrier of (TOP-REAL n)
g5 is set
p1 . g5 is set
g7 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1)) : not p2 <= |.(b₁ - U2).| } is set
g8 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
g8 - U2 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
g8 - U2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(g8 - U2).| is V21() real ext-real non negative Element of REAL
sqr (g8 - U2) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g8 - U2)) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g8 - U2))) is V21() real ext-real Element of REAL
g6 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
g6 - g4 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
g6 - g4 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
p1 . (g6 - g4) is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
g7 - U2 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
g7 - U2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(g6 - g4).| is V21() real ext-real non negative Element of REAL
sqr (g6 - g4) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g6 - g4)) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g6 - g4))) is V21() real ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : not p2 <= |.(b₁ - g4).| } is set
([#] ((n + 1),M,(0. (TOP-REAL (n + 1))))) \ (p1 .: U1) is Element of bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
the topology of ((n + 1),M,(0. (TOP-REAL (n + 1)))) is non empty open Element of bool (bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1)))))
bool (bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))) is non empty set
U is functional Element of bool the carrier of (TOP-REAL (n + 1))
U /\ ([#] ((n + 1),M,(0. (TOP-REAL (n + 1))))) is Element of bool the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
m is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p1 . m is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
S is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
U11 is V21() real ext-real set
Ball (S,U11) is functional open V199( TOP-REAL (n + 1)) V362( TOP-REAL (n + 1)) Element of bool the carrier of (TOP-REAL (n + 1))
Ball (m,U11) is functional open V362( TOP-REAL n) Element of bool the carrier of (TOP-REAL n)
V is set
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : not U11 <= |.(b₁ - m).| } is set
U2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 - m is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 - m is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(U2 - m).| is V21() real ext-real non negative Element of REAL
sqr (U2 - m) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (U2 - m)) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (U2 - m))) is V21() real ext-real Element of REAL
p1 . U2 is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
p1 . (U2 - m) is Element of the carrier of ((n + 1),M,(0. (TOP-REAL (n + 1))))
p2 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
p2 - S is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
p2 - S is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(p2 - S).| is V21() real ext-real non negative Element of REAL
sqr (p2 - S) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (p2 - S)) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (p2 - S))) is V21() real ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1)) : not U11 <= |.(b₁ - S).| } is set
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
n + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
TOP-REAL (n + 1) is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n + 1 -locally_euclidean n + 1 -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL (n + 1)) is functional non empty set
0. (TOP-REAL (n + 1)) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like V51( TOP-REAL (n + 1)) complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
the ZeroF of (TOP-REAL (n + 1)) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
p is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
n1 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
((n + 1),p,n1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL (n + 1)
((n + 1),p,n1) is functional Element of bool the carrier of (TOP-REAL (n + 1))
bool the carrier of (TOP-REAL (n + 1)) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1)) : |(p,(b₁ - n1))| = {} } is set
(TOP-REAL (n + 1)) | ((n + 1),p,n1) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL (n + 1)
Base_FinSeq ((n + 1),(n + 1)) is Relation-like NAT -defined REAL -valued Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of REAL (n + 1)
REAL (n + 1) is functional non empty FinSequence-membered FinSequenceSet of REAL
(n + 1) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
p1 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
{} + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
|.p1.| is V21() real ext-real non negative Element of REAL
sqr p1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr p1) is V21() real ext-real Element of REAL
sqrt (Sum (sqr p1)) is V21() real ext-real Element of REAL
((n + 1),p1,(0. (TOP-REAL (n + 1)))) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL (n + 1)
((n + 1),p1,(0. (TOP-REAL (n + 1)))) is functional Element of bool the carrier of (TOP-REAL (n + 1))
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1)) : |(p1,(b₁ - (0. (TOP-REAL (n + 1)))))| = {} } is set
(TOP-REAL (n + 1)) | ((n + 1),p1,(0. (TOP-REAL (n + 1)))) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL (n + 1)
[: the carrier of (TOP-REAL (n + 1)), the carrier of (TOP-REAL (n + 1)):] is Relation-like non empty set
bool [: the carrier of (TOP-REAL (n + 1)), the carrier of (TOP-REAL (n + 1)):] is non empty set
((n + 1),p,(0. (TOP-REAL (n + 1)))) is functional Element of bool the carrier of (TOP-REAL (n + 1))
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1)) : |(p,(b₁ - (0. (TOP-REAL (n + 1)))))| = {} } is set
A is Relation-like the carrier of (TOP-REAL (n + 1)) -defined the carrier of (TOP-REAL (n + 1)) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (TOP-REAL (n + 1)), the carrier of (TOP-REAL (n + 1)):]
A .: ((n + 1),p1,(0. (TOP-REAL (n + 1)))) is functional Element of bool the carrier of (TOP-REAL (n + 1))
((n + 1),p,(0. (TOP-REAL (n + 1)))) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL (n + 1)
(TOP-REAL (n + 1)) | ((n + 1),p,(0. (TOP-REAL (n + 1)))) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL (n + 1)
transl (n1,(TOP-REAL (n + 1))) is Relation-like the carrier of (TOP-REAL (n + 1)) -defined the carrier of (TOP-REAL (n + 1)) -valued Function-like non empty total quasi_total being_homeomorphism Element of bool [: the carrier of (TOP-REAL (n + 1)), the carrier of (TOP-REAL (n + 1)):]
(transl (n1,(TOP-REAL (n + 1)))) .: ((n + 1),p,(0. (TOP-REAL (n + 1)))) is functional Element of bool the carrier of (TOP-REAL (n + 1))
(0. (TOP-REAL (n + 1))) + n1 is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
(0. (TOP-REAL (n + 1))) + n1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((n + 1),p,((0. (TOP-REAL (n + 1))) + n1)) is functional Element of bool the carrier of (TOP-REAL (n + 1))
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1)) : |(p,(b₁ - ((0. (TOP-REAL (n + 1))) + n1)))| = {} } is set
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
n + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
TOP-REAL (n + 1) is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n + 1 -locally_euclidean n + 1 -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL (n + 1)) is functional non empty set
0. (TOP-REAL (n + 1)) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like V51( TOP-REAL (n + 1)) complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
the ZeroF of (TOP-REAL (n + 1)) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
M is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
p is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
((n + 1),M,p) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL (n + 1)
((n + 1),M,p) is functional Element of bool the carrier of (TOP-REAL (n + 1))
bool the carrier of (TOP-REAL (n + 1)) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1)) : |(M,(b₁ - p))| = {} } is set
(TOP-REAL (n + 1)) | ((n + 1),M,p) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL (n + 1)
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
n + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
Tunit_circle (n + 1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL (n + 1)
TOP-REAL (n + 1) is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n + 1 -locally_euclidean n + 1 -manifold manifold-like RLTopStruct
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
(n) is TopSpace-like TopStruct
n + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
Tunit_circle (n + 1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL (n + 1)
TOP-REAL (n + 1) is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n + 1 -locally_euclidean n + 1 -manifold manifold-like RLTopStruct
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
Sphere ((0. (TOP-REAL n)),1) is functional closed Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
p is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL n
the carrier of p is set
(n,M,(0. (TOP-REAL n))) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL n
(n,M,(0. (TOP-REAL n))) is functional Element of bool the carrier of (TOP-REAL n)
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : |(M,(b₁ - (0. (TOP-REAL n))))| = {} } is set
(TOP-REAL n) | (n,M,(0. (TOP-REAL n))) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL n
the carrier of (n,M,(0. (TOP-REAL n))) is non empty set
[: the carrier of p, the carrier of (n,M,(0. (TOP-REAL n))):] is Relation-like set
bool [: the carrier of p, the carrier of (n,M,(0. (TOP-REAL n))):] is non empty set
[#] p is non proper open closed Element of bool the carrier of p
bool the carrier of p is non empty set
[#] (n,M,(0. (TOP-REAL n))) is non empty non proper open closed Element of bool the carrier of (n,M,(0. (TOP-REAL n)))
bool the carrier of (n,M,(0. (TOP-REAL n))) is non empty set
p1 is set
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
A is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(A,M)| is V21() real ext-real Element of REAL
|(A,M)| * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(A,M)| * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
A - (|(A,M)| * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
A - (|(A,M)| * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 - |(A,M)| is V21() real ext-real Element of REAL
1 / (1 - |(A,M)|) is V21() real ext-real Element of REAL
(1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
TOP-REAL U is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional U -locally_euclidean U -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL U) is functional non empty set
m is Relation-like NAT -defined Function-like finite U -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL U)
0. (TOP-REAL U) is Relation-like NAT -defined Function-like finite U -element FinSequence-like FinSubsequence-like V51( TOP-REAL U) complex-yielding V117() V118() Element of the carrier of (TOP-REAL U)
the ZeroF of (TOP-REAL U) is Relation-like NAT -defined Function-like finite U -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL U)
m - (0. (TOP-REAL U)) is Relation-like NAT -defined Function-like finite U -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL U)
m - (0. (TOP-REAL U)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(m - (0. (TOP-REAL U))).| is V21() real ext-real non negative Element of REAL
sqr (m - (0. (TOP-REAL U))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (m - (0. (TOP-REAL U)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (m - (0. (TOP-REAL U))))) is V21() real ext-real Element of REAL
- (0. (TOP-REAL U)) is Relation-like NAT -defined Function-like finite U -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL U)
- (0. (TOP-REAL U)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
m + (- (0. (TOP-REAL U))) is Relation-like NAT -defined Function-like finite U -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL U)
m + (- (0. (TOP-REAL U))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(m + (- (0. (TOP-REAL U)))).| is V21() real ext-real non negative Element of REAL
sqr (m + (- (0. (TOP-REAL U)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (m + (- (0. (TOP-REAL U))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (m + (- (0. (TOP-REAL U)))))) is V21() real ext-real Element of REAL
- 1 is non empty V21() real ext-real non positive negative Element of REAL
(- 1) * (0. (TOP-REAL U)) is Relation-like NAT -defined Function-like finite U -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL U)
(- 1) * (0. (TOP-REAL U)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
m + ((- 1) * (0. (TOP-REAL U))) is Relation-like NAT -defined Function-like finite U -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL U)
m + ((- 1) * (0. (TOP-REAL U))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(m + ((- 1) * (0. (TOP-REAL U)))).| is V21() real ext-real non negative Element of REAL
sqr (m + ((- 1) * (0. (TOP-REAL U)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (m + ((- 1) * (0. (TOP-REAL U))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (m + ((- 1) * (0. (TOP-REAL U)))))) is V21() real ext-real Element of REAL
m + (0. (TOP-REAL U)) is Relation-like NAT -defined Function-like finite U -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL U)
m + (0. (TOP-REAL U)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(m + (0. (TOP-REAL U))).| is V21() real ext-real non negative Element of REAL
sqr (m + (0. (TOP-REAL U))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (m + (0. (TOP-REAL U)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (m + (0. (TOP-REAL U))))) is V21() real ext-real Element of REAL
|.M.| is V21() real ext-real non negative Element of REAL
sqr M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr M) is V21() real ext-real Element of REAL
sqrt (Sum (sqr M)) is V21() real ext-real Element of REAL
|(M,M)| is V21() real ext-real Element of REAL
1 ^2 is V21() real ext-real Element of REAL
1 * 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
|(M,(|(A,M)| * M))| is V21() real ext-real Element of REAL
|(A,M)| * |(M,M)| is V21() real ext-real Element of REAL
|(M,A)| is V21() real ext-real Element of REAL
|(M,((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))))| is V21() real ext-real Element of REAL
|(M,(A - (|(A,M)| * M)))| is V21() real ext-real Element of REAL
(1 / (1 - |(A,M)|)) * |(M,(A - (|(A,M)| * M)))| is V21() real ext-real Element of REAL
|(M,A)| - |(M,(|(A,M)| * M))| is V21() real ext-real Element of REAL
(1 / (1 - |(A,M)|)) * (|(M,A)| - |(M,(|(A,M)| * M))|) is V21() real ext-real Element of REAL
((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) + (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) + (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) + (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
(- 1) * (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(- 1) * (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) + ((- 1) * (0. (TOP-REAL n)))))| is V21() real ext-real Element of REAL
- (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
- (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) + (- (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) + (- (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) + (- (0. (TOP-REAL n)))))| is V21() real ext-real Element of REAL
((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(((1 / (1 - |(A,M)|)) * (A - (|(A,M)| * M))) - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
S is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(S,M)| is V21() real ext-real Element of REAL
|(S,M)| * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(S,M)| * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
S - (|(S,M)| * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
S - (|(S,M)| * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 - |(S,M)| is V21() real ext-real Element of REAL
1 / (1 - |(S,M)|) is V21() real ext-real Element of REAL
(1 / (1 - |(S,M)|)) * (S - (|(S,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(S,M)|)) * (S - (|(S,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
[:([#] p),([#] (n,M,(0. (TOP-REAL n)))):] is Relation-like set
bool [:([#] p),([#] (n,M,(0. (TOP-REAL n)))):] is non empty set
p1 is Relation-like [#] p -defined [#] (n,M,(0. (TOP-REAL n))) -valued Function-like total quasi_total Element of bool [:([#] p),([#] (n,M,(0. (TOP-REAL n)))):]
A is Relation-like the carrier of p -defined the carrier of (n,M,(0. (TOP-REAL n))) -valued Function-like total quasi_total Element of bool [: the carrier of p, the carrier of (n,M,(0. (TOP-REAL n))):]
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
A . U1 is set
|(U1,M)| is V21() real ext-real Element of REAL
|(U1,M)| * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(U1,M)| * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U1 - (|(U1,M)| * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 - (|(U1,M)| * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 - |(U1,M)| is V21() real ext-real Element of REAL
1 / (1 - |(U1,M)|) is V21() real ext-real Element of REAL
(1 / (1 - |(U1,M)|)) * (U1 - (|(U1,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(U1,M)|)) * (U1 - (|(U1,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
n1 is Relation-like the carrier of p -defined the carrier of (n,M,(0. (TOP-REAL n))) -valued Function-like total quasi_total Element of bool [: the carrier of p, the carrier of (n,M,(0. (TOP-REAL n))):]
p1 is Relation-like the carrier of p -defined the carrier of (n,M,(0. (TOP-REAL n))) -valued Function-like total quasi_total Element of bool [: the carrier of p, the carrier of (n,M,(0. (TOP-REAL n))):]
[#] p is non proper open closed Element of bool the carrier of p
bool the carrier of p is non empty set
A is set
n1 . A is set
p1 . A is set
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
U1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(U1,M)| is V21() real ext-real Element of REAL
|(U1,M)| * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(U1,M)| * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U1 - (|(U1,M)| * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U1 - (|(U1,M)| * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 - |(U1,M)| is V21() real ext-real Element of REAL
1 / (1 - |(U1,M)|) is V21() real ext-real Element of REAL
(1 / (1 - |(U1,M)|)) * (U1 - (|(U1,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(U1,M)|)) * (U1 - (|(U1,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
0. (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like V51( TOP-REAL n) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
Sphere ((0. (TOP-REAL n)),1) is functional closed Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
{M} is functional non empty finite V36() set
(Sphere ((0. (TOP-REAL n)),1)) \ {M} is functional Element of bool the carrier of (TOP-REAL n)
(n,M,(0. (TOP-REAL n))) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL n
(n,M,(0. (TOP-REAL n))) is functional Element of bool the carrier of (TOP-REAL n)
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n) : |(M,(b₁ - (0. (TOP-REAL n))))| = {} } is set
(TOP-REAL n) | (n,M,(0. (TOP-REAL n))) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL n
p is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL n
[#] p is non proper open closed Element of bool the carrier of p
the carrier of p is set
bool the carrier of p is non empty set
(n,M,p) is Relation-like the carrier of p -defined the carrier of (n,M,(0. (TOP-REAL n))) -valued Function-like total quasi_total Element of bool [: the carrier of p, the carrier of (n,M,(0. (TOP-REAL n))):]
the carrier of (n,M,(0. (TOP-REAL n))) is non empty set
[: the carrier of p, the carrier of (n,M,(0. (TOP-REAL n))):] is Relation-like set
bool [: the carrier of p, the carrier of (n,M,(0. (TOP-REAL n))):] is non empty set
dom (n,M,p) is Element of bool the carrier of p
A is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
TOP-REAL A is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional A -locally_euclidean A -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL A) is functional non empty set
U1 is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
0. (TOP-REAL A) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like V51( TOP-REAL A) complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
the ZeroF of (TOP-REAL A) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
U1 - (0. (TOP-REAL A)) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
U1 - (0. (TOP-REAL A)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(U1 - (0. (TOP-REAL A))).| is V21() real ext-real non negative Element of REAL
sqr (U1 - (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (U1 - (0. (TOP-REAL A)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (U1 - (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
- (0. (TOP-REAL A)) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
- (0. (TOP-REAL A)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U1 + (- (0. (TOP-REAL A))) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
U1 + (- (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(U1 + (- (0. (TOP-REAL A)))).| is V21() real ext-real non negative Element of REAL
sqr (U1 + (- (0. (TOP-REAL A)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (U1 + (- (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (U1 + (- (0. (TOP-REAL A)))))) is V21() real ext-real Element of REAL
- 1 is non empty V21() real ext-real non positive negative Element of REAL
(- 1) * (0. (TOP-REAL A)) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
(- 1) * (0. (TOP-REAL A)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U1 + ((- 1) * (0. (TOP-REAL A))) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
U1 + ((- 1) * (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(U1 + ((- 1) * (0. (TOP-REAL A)))).| is V21() real ext-real non negative Element of REAL
sqr (U1 + ((- 1) * (0. (TOP-REAL A)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (U1 + ((- 1) * (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (U1 + ((- 1) * (0. (TOP-REAL A)))))) is V21() real ext-real Element of REAL
U1 + (0. (TOP-REAL A)) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
U1 + (0. (TOP-REAL A)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(U1 + (0. (TOP-REAL A))).| is V21() real ext-real non negative Element of REAL
sqr (U1 + (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (U1 + (0. (TOP-REAL A)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (U1 + (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
|.M.| is V21() real ext-real non negative Element of REAL
sqr M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr M) is V21() real ext-real Element of REAL
sqrt (Sum (sqr M)) is V21() real ext-real Element of REAL
|(M,M)| is V21() real ext-real Element of REAL
1 ^2 is V21() real ext-real Element of REAL
1 * 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
[#] (n,M,(0. (TOP-REAL n))) is non empty non proper open closed Element of bool the carrier of (n,M,(0. (TOP-REAL n)))
bool the carrier of (n,M,(0. (TOP-REAL n))) is non empty set
U is set
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
m is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
2 * m is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
2 * m is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(m,m)| is V21() real ext-real Element of REAL
|(m,m)| - 1 is V21() real ext-real Element of REAL
(|(m,m)| - 1) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(|(m,m)| - 1) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(2 * m) + ((|(m,m)| - 1) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(2 * m) + ((|(m,m)| - 1) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(m,m)| + 1 is V21() real ext-real Element of REAL
1 / (|(m,m)| + 1) is V21() real ext-real Element of REAL
(1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U11 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
TOP-REAL U11 is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional U11 -locally_euclidean U11 -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL U11) is functional non empty set
U2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(U2 - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
- (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
- (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
m + (- (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
m + (- (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(m + (- (0. (TOP-REAL n)))))| is V21() real ext-real Element of REAL
(- 1) * (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(- 1) * (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
m + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
m + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(m + ((- 1) * (0. (TOP-REAL n)))))| is V21() real ext-real Element of REAL
m + (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
m + (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(m + (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
|(M,m)| is V21() real ext-real Element of REAL
|(((2 * m) + ((|(m,m)| - 1) * M)),M)| is V21() real ext-real Element of REAL
|(m,M)| is V21() real ext-real Element of REAL
2 * |(m,M)| is V21() real ext-real Element of REAL
(|(m,m)| - 1) * |(M,M)| is V21() real ext-real Element of REAL
(2 * |(m,M)|) + ((|(m,m)| - 1) * |(M,M)|) is V21() real ext-real Element of REAL
|(((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))),M)| is V21() real ext-real Element of REAL
(1 / (|(m,m)| + 1)) * |(((2 * m) + ((|(m,m)| - 1) * M)),M)| is V21() real ext-real Element of REAL
(1 / (|(m,m)| + 1)) * (|(m,m)| - 1) is V21() real ext-real Element of REAL
((1 / (|(m,m)| + 1)) * (|(m,m)| - 1)) * |(M,M)| is V21() real ext-real Element of REAL
(|(m,m)| + 1) * 1 is V21() real ext-real Element of REAL
(|(m,m)| + 1) * ((1 / (|(m,m)| + 1)) * (|(m,m)| - 1)) is V21() real ext-real Element of REAL
(|(m,m)| + 1) * (1 / (|(m,m)| + 1)) is V21() real ext-real Element of REAL
((|(m,m)| + 1) * (1 / (|(m,m)| + 1))) * (|(m,m)| - 1) is V21() real ext-real Element of REAL
(|(m,m)| + 1) / (|(m,m)| + 1) is V21() real ext-real Element of REAL
((|(m,m)| + 1) / (|(m,m)| + 1)) * (|(m,m)| - 1) is V21() real ext-real Element of REAL
1 * (|(m,m)| - 1) is V21() real ext-real Element of REAL
|(m,((2 * m) + ((|(m,m)| - 1) * M)))| is V21() real ext-real Element of REAL
2 * |(m,m)| is V21() real ext-real Element of REAL
(|(m,m)| - 1) * |(M,m)| is V21() real ext-real Element of REAL
(2 * |(m,m)|) + ((|(m,m)| - 1) * |(M,m)|) is V21() real ext-real Element of REAL
|(M,((2 * m) + ((|(m,m)| - 1) * M)))| is V21() real ext-real Element of REAL
|(m,M)| is V21() real ext-real Element of REAL
2 * |(m,M)| is V21() real ext-real Element of REAL
(|(m,m)| - 1) * |(M,M)| is V21() real ext-real Element of REAL
(2 * |(m,M)|) + ((|(m,m)| - 1) * |(M,M)|) is V21() real ext-real Element of REAL
|(((2 * m) + ((|(m,m)| - 1) * M)),((2 * m) + ((|(m,m)| - 1) * M)))| is V21() real ext-real Element of REAL
2 * (2 * |(m,m)|) is V21() real ext-real Element of REAL
(|(m,m)| - 1) * |(M,((2 * m) + ((|(m,m)| - 1) * M)))| is V21() real ext-real Element of REAL
(2 * (2 * |(m,m)|)) + ((|(m,m)| - 1) * |(M,((2 * m) + ((|(m,m)| - 1) * M)))|) is V21() real ext-real Element of REAL
(|(m,m)| + 1) * (|(m,m)| + 1) is V21() real ext-real Element of REAL
|(((2 * m) + ((|(m,m)| - 1) * M)),((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))))| is V21() real ext-real Element of REAL
(1 / (|(m,m)| + 1)) * |(((2 * m) + ((|(m,m)| - 1) * M)),((2 * m) + ((|(m,m)| - 1) * M)))| is V21() real ext-real Element of REAL
|(((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))),((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))))| is V21() real ext-real Element of REAL
(1 / (|(m,m)| + 1)) * |(((2 * m) + ((|(m,m)| - 1) * M)),((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))))| is V21() real ext-real Element of REAL
(1 / (|(m,m)| + 1)) * (|(m,m)| + 1) is V21() real ext-real Element of REAL
(1 / (|(m,m)| + 1)) * ((1 / (|(m,m)| + 1)) * (|(m,m)| + 1)) is V21() real ext-real Element of REAL
((1 / (|(m,m)| + 1)) * ((1 / (|(m,m)| + 1)) * (|(m,m)| + 1))) * (|(m,m)| + 1) is V21() real ext-real Element of REAL
(|(m,m)| + 1) / (|(m,m)| + 1) is V21() real ext-real Element of REAL
(1 / (|(m,m)| + 1)) * ((|(m,m)| + 1) / (|(m,m)| + 1)) is V21() real ext-real Element of REAL
((1 / (|(m,m)| + 1)) * ((|(m,m)| + 1) / (|(m,m)| + 1))) * (|(m,m)| + 1) is V21() real ext-real Element of REAL
(1 / (|(m,m)| + 1)) * 1 is V21() real ext-real Element of REAL
((1 / (|(m,m)| + 1)) * 1) * (|(m,m)| + 1) is V21() real ext-real Element of REAL
|.((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))).| is V21() real ext-real non negative Element of REAL
sqr ((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr ((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr ((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))))) is V21() real ext-real Element of REAL
((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (0. (TOP-REAL n))).| is V21() real ext-real non negative Element of REAL
sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (0. (TOP-REAL n)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (0. (TOP-REAL n))))) is V21() real ext-real Element of REAL
((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + ((- 1) * (0. (TOP-REAL n)))).| is V21() real ext-real non negative Element of REAL
sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + ((- 1) * (0. (TOP-REAL n)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + ((- 1) * (0. (TOP-REAL n))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + ((- 1) * (0. (TOP-REAL n)))))) is V21() real ext-real Element of REAL
((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (- (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (- (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (- (0. (TOP-REAL n)))).| is V21() real ext-real non negative Element of REAL
sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (- (0. (TOP-REAL n)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (- (0. (TOP-REAL n))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) + (- (0. (TOP-REAL n)))))) is V21() real ext-real Element of REAL
((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) - (0. (TOP-REAL n))).| is V21() real ext-real non negative Element of REAL
sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) - (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) - (0. (TOP-REAL n)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (((1 / (|(m,m)| + 1)) * ((2 * m) + ((|(m,m)| - 1) * M))) - (0. (TOP-REAL n))))) is V21() real ext-real Element of REAL
p2 is Relation-like NAT -defined Function-like finite U11 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL U11)
g4 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
2 * g4 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
2 * g4 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(g4,g4)| is V21() real ext-real Element of REAL
|(g4,g4)| - 1 is V21() real ext-real Element of REAL
(|(g4,g4)| - 1) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(|(g4,g4)| - 1) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(2 * g4) + ((|(g4,g4)| - 1) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(2 * g4) + ((|(g4,g4)| - 1) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(g4,g4)| + 1 is V21() real ext-real Element of REAL
1 / (|(g4,g4)| + 1) is V21() real ext-real Element of REAL
(1 / (|(g4,g4)| + 1)) * ((2 * g4) + ((|(g4,g4)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (|(g4,g4)| + 1)) * ((2 * g4) + ((|(g4,g4)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
[:([#] (n,M,(0. (TOP-REAL n)))),([#] p):] is Relation-like set
bool [:([#] (n,M,(0. (TOP-REAL n)))),([#] p):] is non empty set
U is Relation-like [#] (n,M,(0. (TOP-REAL n))) -defined [#] p -valued Function-like quasi_total Element of bool [:([#] (n,M,(0. (TOP-REAL n)))),([#] p):]
[: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of p:] is Relation-like set
bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of p:] is non empty set
[:([#] p),([#] (n,M,(0. (TOP-REAL n)))):] is Relation-like set
bool [:([#] p),([#] (n,M,(0. (TOP-REAL n)))):] is non empty set
|.(0. (TOP-REAL n)).| is V21() real ext-real non negative Element of REAL
sqr (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (0. (TOP-REAL n))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (0. (TOP-REAL n)))) is V21() real ext-real Element of REAL
1 + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
(1 + 1) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 + 1) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
1 * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 * M) + (1 * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 * M) + (1 * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 * M) + M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 * M) + M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
M + M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
M + M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
- M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
- M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
M + (- M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
M + (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(- M).| is V21() real ext-real non negative Element of REAL
sqr (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (- M)) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (- M))) is V21() real ext-real Element of REAL
(- M) + (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(- M) + (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.((- M) + (0. (TOP-REAL n))).| is V21() real ext-real non negative Element of REAL
sqr ((- M) + (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr ((- M) + (0. (TOP-REAL n)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr ((- M) + (0. (TOP-REAL n))))) is V21() real ext-real Element of REAL
(- 1) * (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(- 1) * (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(- M) + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(- M) + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.((- M) + ((- 1) * (0. (TOP-REAL n)))).| is V21() real ext-real non negative Element of REAL
sqr ((- M) + ((- 1) * (0. (TOP-REAL n)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr ((- M) + ((- 1) * (0. (TOP-REAL n))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr ((- M) + ((- 1) * (0. (TOP-REAL n)))))) is V21() real ext-real Element of REAL
- (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
- (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(- M) + (- (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(- M) + (- (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.((- M) + (- (0. (TOP-REAL n)))).| is V21() real ext-real non negative Element of REAL
sqr ((- M) + (- (0. (TOP-REAL n)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr ((- M) + (- (0. (TOP-REAL n))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr ((- M) + (- (0. (TOP-REAL n)))))) is V21() real ext-real Element of REAL
(- M) - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(- M) - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.((- M) - (0. (TOP-REAL n))).| is V21() real ext-real non negative Element of REAL
sqr ((- M) - (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr ((- M) - (0. (TOP-REAL n)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr ((- M) - (0. (TOP-REAL n))))) is V21() real ext-real Element of REAL
Sphere ((0. (TOP-REAL A)),1) is functional closed V199( TOP-REAL A) Element of bool the carrier of (TOP-REAL A)
bool the carrier of (TOP-REAL A) is non empty set
S is Relation-like [#] p -defined [#] (n,M,(0. (TOP-REAL n))) -valued Function-like total quasi_total Element of bool [:([#] p),([#] (n,M,(0. (TOP-REAL n)))):]
U11 is set
U . U11 is set
V is set
S . V is set
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
p2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
2 * p2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
2 * p2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(p2,p2)| is V21() real ext-real Element of REAL
|(p2,p2)| - 1 is V21() real ext-real Element of REAL
(|(p2,p2)| - 1) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(|(p2,p2)| - 1) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(2 * p2) + ((|(p2,p2)| - 1) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(2 * p2) + ((|(p2,p2)| - 1) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(p2,p2)| + 1 is V21() real ext-real Element of REAL
1 / (|(p2,p2)| + 1) is V21() real ext-real Element of REAL
(1 / (|(p2,p2)| + 1)) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (|(p2,p2)| + 1)) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(n,M,p) . U2 is set
|(U2,M)| is V21() real ext-real Element of REAL
|(U2,M)| * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(U2,M)| * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U2 - (|(U2,M)| * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 - (|(U2,M)| * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 - |(U2,M)| is V21() real ext-real Element of REAL
1 / (1 - |(U2,M)|) is V21() real ext-real Element of REAL
(1 / (1 - |(U2,M)|)) * (U2 - (|(U2,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(U2,M)|)) * (U2 - (|(U2,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
g4 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
g4 - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
g4 - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(g4 - (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
p2 + (- (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p2 + (- (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(p2 + (- (0. (TOP-REAL n)))))| is V21() real ext-real Element of REAL
p2 + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p2 + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(p2 + ((- 1) * (0. (TOP-REAL n)))))| is V21() real ext-real Element of REAL
p2 + (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p2 + (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(M,(p2 + (0. (TOP-REAL n))))| is V21() real ext-real Element of REAL
|(M,p2)| is V21() real ext-real Element of REAL
|(((2 * p2) + ((|(p2,p2)| - 1) * M)),M)| is V21() real ext-real Element of REAL
|(p2,M)| is V21() real ext-real Element of REAL
2 * |(p2,M)| is V21() real ext-real Element of REAL
(|(p2,p2)| - 1) * |(M,M)| is V21() real ext-real Element of REAL
(2 * |(p2,M)|) + ((|(p2,p2)| - 1) * |(M,M)|) is V21() real ext-real Element of REAL
(1 / (|(p2,p2)| + 1)) * |(((2 * p2) + ((|(p2,p2)| - 1) * M)),M)| is V21() real ext-real Element of REAL
(|(p2,p2)| - 1) / (|(p2,p2)| + 1) is V21() real ext-real Element of REAL
((|(p2,p2)| - 1) / (|(p2,p2)| + 1)) * 1 is V21() real ext-real Element of REAL
(|(p2,p2)| + 1) / (|(p2,p2)| + 1) is V21() real ext-real Element of REAL
((|(p2,p2)| + 1) / (|(p2,p2)| + 1)) - ((|(p2,p2)| - 1) / (|(p2,p2)| + 1)) is V21() real ext-real Element of REAL
(|(p2,p2)| + 1) - (|(p2,p2)| - 1) is V21() real ext-real Element of REAL
((|(p2,p2)| + 1) - (|(p2,p2)| - 1)) / (|(p2,p2)| + 1) is V21() real ext-real Element of REAL
2 / (|(p2,p2)| + 1) is V21() real ext-real Element of REAL
(|(p2,p2)| + 1) / 2 is V21() real ext-real Element of REAL
((|(p2,p2)| + 1) / 2) * U2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((|(p2,p2)| + 1) / 2) * U2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((|(p2,p2)| + 1) / 2) * (1 / (|(p2,p2)| + 1)) is V21() real ext-real Element of REAL
(((|(p2,p2)| + 1) / 2) * (1 / (|(p2,p2)| + 1))) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((|(p2,p2)| + 1) / 2) * (1 / (|(p2,p2)| + 1))) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(|(p2,p2)| + 1) * 1 is V21() real ext-real Element of REAL
2 * (|(p2,p2)| + 1) is V21() real ext-real Element of REAL
((|(p2,p2)| + 1) * 1) / (2 * (|(p2,p2)| + 1)) is V21() real ext-real Element of REAL
(((|(p2,p2)| + 1) * 1) / (2 * (|(p2,p2)| + 1))) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((|(p2,p2)| + 1) * 1) / (2 * (|(p2,p2)| + 1))) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 / 2 is non empty V21() real ext-real positive non negative Element of REAL
((|(p2,p2)| + 1) / (|(p2,p2)| + 1)) * (1 / 2) is V21() real ext-real Element of REAL
(((|(p2,p2)| + 1) / (|(p2,p2)| + 1)) * (1 / 2)) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((|(p2,p2)| + 1) / (|(p2,p2)| + 1)) * (1 / 2)) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 * (1 / 2) is non empty V21() real ext-real positive non negative Element of REAL
(1 * (1 / 2)) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 * (1 / 2)) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / 2) * (2 * p2) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / 2) * (2 * p2) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / 2) * ((|(p2,p2)| - 1) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / 2) * ((|(p2,p2)| - 1) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((1 / 2) * (2 * p2)) + ((1 / 2) * ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / 2) * (2 * p2)) + ((1 / 2) * ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / 2) * 2 is non empty V21() real ext-real positive non negative Element of REAL
((1 / 2) * 2) * p2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / 2) * 2) * p2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(((1 / 2) * 2) * p2) + ((1 / 2) * ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((1 / 2) * 2) * p2) + ((1 / 2) * ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 * p2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
1 * p2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / 2) * (|(p2,p2)| - 1) is V21() real ext-real Element of REAL
((1 / 2) * (|(p2,p2)| - 1)) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / 2) * (|(p2,p2)| - 1)) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 * p2) + (((1 / 2) * (|(p2,p2)| - 1)) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 * p2) + (((1 / 2) * (|(p2,p2)| - 1)) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(|(p2,p2)| - 1) / 2 is V21() real ext-real Element of REAL
((|(p2,p2)| - 1) / 2) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((|(p2,p2)| - 1) / 2) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
p2 + (((|(p2,p2)| - 1) / 2) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
p2 + (((|(p2,p2)| - 1) / 2) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((|(p2,p2)| + 1) / 2) * |(U2,M)| is V21() real ext-real Element of REAL
((|(p2,p2)| + 1) / (|(p2,p2)| + 1)) * ((|(p2,p2)| - 1) / 2) is V21() real ext-real Element of REAL
1 * ((|(p2,p2)| - 1) / 2) is V21() real ext-real Element of REAL
((|(p2,p2)| + 1) / 2) * (|(U2,M)| * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((|(p2,p2)| + 1) / 2) * (|(U2,M)| * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(((|(p2,p2)| + 1) / 2) * U2) - (((|(p2,p2)| + 1) / 2) * (|(U2,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((|(p2,p2)| + 1) / 2) * U2) - (((|(p2,p2)| + 1) / 2) * (|(U2,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(((|(p2,p2)| + 1) / 2) * U2) - (((|(p2,p2)| - 1) / 2) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((|(p2,p2)| + 1) / 2) * U2) - (((|(p2,p2)| - 1) / 2) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(U2,M)| is V21() real ext-real Element of REAL
|(U2,M)| * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(U2,M)| * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U2 - (|(U2,M)| * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 - (|(U2,M)| * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 - |(U2,M)| is V21() real ext-real Element of REAL
1 / (1 - |(U2,M)|) is V21() real ext-real Element of REAL
(1 / (1 - |(U2,M)|)) * (U2 - (|(U2,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(U2,M)|)) * (U2 - (|(U2,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
p2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U . p2 is set
2 * p2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
2 * p2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(p2,p2)| is V21() real ext-real Element of REAL
|(p2,p2)| - 1 is V21() real ext-real Element of REAL
(|(p2,p2)| - 1) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(|(p2,p2)| - 1) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(2 * p2) + ((|(p2,p2)| - 1) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(2 * p2) + ((|(p2,p2)| - 1) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(p2,p2)| + 1 is V21() real ext-real Element of REAL
1 / (|(p2,p2)| + 1) is V21() real ext-real Element of REAL
(1 / (|(p2,p2)| + 1)) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (|(p2,p2)| + 1)) * ((2 * p2) + ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
g4 is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g4 - (0. (TOP-REAL A)) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g4 - (0. (TOP-REAL A)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(g4 - (0. (TOP-REAL A))).| is V21() real ext-real non negative Element of REAL
sqr (g4 - (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g4 - (0. (TOP-REAL A)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g4 - (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
g4 + (- (0. (TOP-REAL A))) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g4 + (- (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(g4 + (- (0. (TOP-REAL A)))).| is V21() real ext-real non negative Element of REAL
sqr (g4 + (- (0. (TOP-REAL A)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g4 + (- (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g4 + (- (0. (TOP-REAL A)))))) is V21() real ext-real Element of REAL
g4 + ((- 1) * (0. (TOP-REAL A))) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g4 + ((- 1) * (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(g4 + ((- 1) * (0. (TOP-REAL A)))).| is V21() real ext-real non negative Element of REAL
sqr (g4 + ((- 1) * (0. (TOP-REAL A)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g4 + ((- 1) * (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g4 + ((- 1) * (0. (TOP-REAL A)))))) is V21() real ext-real Element of REAL
g4 + (0. (TOP-REAL A)) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g4 + (0. (TOP-REAL A)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(g4 + (0. (TOP-REAL A))).| is V21() real ext-real non negative Element of REAL
sqr (g4 + (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g4 + (0. (TOP-REAL A)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g4 + (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
|.U2.| is V21() real ext-real non negative Element of REAL
sqr U2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr U2) is V21() real ext-real Element of REAL
sqrt (Sum (sqr U2)) is V21() real ext-real Element of REAL
|(U2,U2)| is V21() real ext-real Element of REAL
|((|(U2,M)| * M),(U2 - (|(U2,M)| * M)))| is V21() real ext-real Element of REAL
|((|(U2,M)| * M),U2)| is V21() real ext-real Element of REAL
|((|(U2,M)| * M),(|(U2,M)| * M))| is V21() real ext-real Element of REAL
|((|(U2,M)| * M),U2)| - |((|(U2,M)| * M),(|(U2,M)| * M))| is V21() real ext-real Element of REAL
|(U2,M)| * |(U2,M)| is V21() real ext-real Element of REAL
(|(U2,M)| * |(U2,M)|) - |((|(U2,M)| * M),(|(U2,M)| * M))| is V21() real ext-real Element of REAL
|(M,(|(U2,M)| * M))| is V21() real ext-real Element of REAL
|(U2,M)| * |(M,(|(U2,M)| * M))| is V21() real ext-real Element of REAL
(|(U2,M)| * |(U2,M)|) - (|(U2,M)| * |(M,(|(U2,M)| * M))|) is V21() real ext-real Element of REAL
|(U2,M)| * |(M,M)| is V21() real ext-real Element of REAL
|(U2,M)| * (|(U2,M)| * |(M,M)|) is V21() real ext-real Element of REAL
(|(U2,M)| * |(U2,M)|) - (|(U2,M)| * (|(U2,M)| * |(M,M)|)) is V21() real ext-real Element of REAL
|(U2,(U2 - (|(U2,M)| * M)))| is V21() real ext-real Element of REAL
|(U2,(|(U2,M)| * M))| is V21() real ext-real Element of REAL
|(U2,U2)| - |(U2,(|(U2,M)| * M))| is V21() real ext-real Element of REAL
1 - (|(U2,M)| * |(U2,M)|) is V21() real ext-real Element of REAL
|((U2 - (|(U2,M)| * M)),(U2 - (|(U2,M)| * M)))| is V21() real ext-real Element of REAL
|(U2,(U2 - (|(U2,M)| * M)))| - |((|(U2,M)| * M),(U2 - (|(U2,M)| * M)))| is V21() real ext-real Element of REAL
(1 - (|(U2,M)| * |(U2,M)|)) + {} is V21() real ext-real Element of REAL
|(U2,M)| - |(U2,U2)| is V21() real ext-real Element of REAL
M - U2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
M - U2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(U2,(M - U2))| is V21() real ext-real Element of REAL
|(M,M)| - |(U2,M)| is V21() real ext-real Element of REAL
|((M - U2),M)| is V21() real ext-real Element of REAL
|((M - U2),(M - U2))| is V21() real ext-real Element of REAL
{} - {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural real finite finite-yielding V36() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding V117() V118() V119() R-orthogonal R-normal R-orthonormal Element of REAL
(1 - |(U2,M)|) * (1 - |(U2,M)|) is V21() real ext-real Element of REAL
|((U2 - (|(U2,M)| * M)),p2)| is V21() real ext-real Element of REAL
(1 / (1 - |(U2,M)|)) * |((U2 - (|(U2,M)| * M)),p2)| is V21() real ext-real Element of REAL
(1 / (1 - |(U2,M)|)) * |((U2 - (|(U2,M)| * M)),(U2 - (|(U2,M)| * M)))| is V21() real ext-real Element of REAL
(1 / (1 - |(U2,M)|)) * ((1 / (1 - |(U2,M)|)) * |((U2 - (|(U2,M)| * M)),(U2 - (|(U2,M)| * M)))|) is V21() real ext-real Element of REAL
(1 / (1 - |(U2,M)|)) * (1 / (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
((1 / (1 - |(U2,M)|)) * (1 / (1 - |(U2,M)|))) * (1 - (|(U2,M)| * |(U2,M)|)) is V21() real ext-real Element of REAL
1 / ((1 - |(U2,M)|) * (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
(1 / ((1 - |(U2,M)|) * (1 - |(U2,M)|))) * (1 - (|(U2,M)| * |(U2,M)|)) is V21() real ext-real Element of REAL
2 * |(U2,M)| is V21() real ext-real Element of REAL
1 - (2 * |(U2,M)|) is V21() real ext-real Element of REAL
(1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|) is V21() real ext-real Element of REAL
(1 - (|(U2,M)| * |(U2,M)|)) / ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|)) is V21() real ext-real Element of REAL
((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|)) / ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|)) is V21() real ext-real Element of REAL
((1 - (|(U2,M)| * |(U2,M)|)) / ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|))) + (((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|)) / ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|))) is V21() real ext-real Element of REAL
(1 - (|(U2,M)| * |(U2,M)|)) + ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|)) is V21() real ext-real Element of REAL
((1 - (|(U2,M)| * |(U2,M)|)) + ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|))) / ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|)) is V21() real ext-real Element of REAL
2 * (1 - |(U2,M)|) is V21() real ext-real Element of REAL
(2 * (1 - |(U2,M)|)) / ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|)) is V21() real ext-real Element of REAL
(1 - |(U2,M)|) / ((1 - |(U2,M)|) * (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
2 * ((1 - |(U2,M)|) / ((1 - |(U2,M)|) * (1 - |(U2,M)|))) is V21() real ext-real Element of REAL
(1 - |(U2,M)|) / (1 - |(U2,M)|) is V21() real ext-real Element of REAL
((1 - |(U2,M)|) / (1 - |(U2,M)|)) / (1 - |(U2,M)|) is V21() real ext-real Element of REAL
2 * (((1 - |(U2,M)|) / (1 - |(U2,M)|)) / (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
2 * (1 / (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
2 * 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
(2 * 1) / (1 - |(U2,M)|) is V21() real ext-real Element of REAL
((1 - (|(U2,M)| * |(U2,M)|)) / ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|))) - (((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|)) / ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|))) is V21() real ext-real Element of REAL
(1 - (|(U2,M)| * |(U2,M)|)) - ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|)) is V21() real ext-real Element of REAL
((1 - (|(U2,M)| * |(U2,M)|)) - ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|))) / ((1 - (2 * |(U2,M)|)) + (|(U2,M)| * |(U2,M)|)) is V21() real ext-real Element of REAL
(2 * |(U2,M)|) * (1 - |(U2,M)|) is V21() real ext-real Element of REAL
((2 * |(U2,M)|) * (1 - |(U2,M)|)) / ((1 - |(U2,M)|) * (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
(2 * |(U2,M)|) * ((1 - |(U2,M)|) / ((1 - |(U2,M)|) * (1 - |(U2,M)|))) is V21() real ext-real Element of REAL
(2 * |(U2,M)|) * (((1 - |(U2,M)|) / (1 - |(U2,M)|)) / (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
(2 * |(U2,M)|) * (1 / (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
(2 * |(U2,M)|) * 1 is V21() real ext-real Element of REAL
((2 * |(U2,M)|) * 1) / (1 - |(U2,M)|) is V21() real ext-real Element of REAL
(1 - |(U2,M)|) / 2 is V21() real ext-real Element of REAL
(|(p2,p2)| - 1) / (|(p2,p2)| + 1) is V21() real ext-real Element of REAL
2 / (1 - |(U2,M)|) is V21() real ext-real Element of REAL
(1 - |(U2,M)|) * (2 / (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
(2 * |(U2,M)|) / ((1 - |(U2,M)|) * (2 / (1 - |(U2,M)|))) is V21() real ext-real Element of REAL
(2 * (1 - |(U2,M)|)) / (1 - |(U2,M)|) is V21() real ext-real Element of REAL
(2 * |(U2,M)|) / ((2 * (1 - |(U2,M)|)) / (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
2 * ((1 - |(U2,M)|) / (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
(2 * |(U2,M)|) / (2 * ((1 - |(U2,M)|) / (1 - |(U2,M)|))) is V21() real ext-real Element of REAL
(2 * |(U2,M)|) / (2 * 1) is V21() real ext-real Element of REAL
(1 - |(U2,M)|) * p2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 - |(U2,M)|) * p2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 - |(U2,M)|) * (1 / (1 - |(U2,M)|)) is V21() real ext-real Element of REAL
((1 - |(U2,M)|) * (1 / (1 - |(U2,M)|))) * (U2 - (|(U2,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 - |(U2,M)|) * (1 / (1 - |(U2,M)|))) * (U2 - (|(U2,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 - |(U2,M)|) * 1 is V21() real ext-real Element of REAL
((1 - |(U2,M)|) * 1) / (1 - |(U2,M)|) is V21() real ext-real Element of REAL
(((1 - |(U2,M)|) * 1) / (1 - |(U2,M)|)) * (U2 - (|(U2,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((1 - |(U2,M)|) * 1) / (1 - |(U2,M)|)) * (U2 - (|(U2,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 * (U2 - (|(U2,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
1 * (U2 - (|(U2,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (|(p2,p2)| + 1)) * (2 * p2) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (|(p2,p2)| + 1)) * (2 * p2) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (|(p2,p2)| + 1)) * ((|(p2,p2)| - 1) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (|(p2,p2)| + 1)) * ((|(p2,p2)| - 1) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((1 / (|(p2,p2)| + 1)) * (2 * p2)) + ((1 / (|(p2,p2)| + 1)) * ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (|(p2,p2)| + 1)) * (2 * p2)) + ((1 / (|(p2,p2)| + 1)) * ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (|(p2,p2)| + 1)) * 2 is V21() real ext-real Element of REAL
((1 / (|(p2,p2)| + 1)) * 2) * p2 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (|(p2,p2)| + 1)) * 2) * p2 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(((1 / (|(p2,p2)| + 1)) * 2) * p2) + ((1 / (|(p2,p2)| + 1)) * ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((1 / (|(p2,p2)| + 1)) * 2) * p2) + ((1 / (|(p2,p2)| + 1)) * ((|(p2,p2)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (|(p2,p2)| + 1)) * (|(p2,p2)| - 1) is V21() real ext-real Element of REAL
((1 / (|(p2,p2)| + 1)) * (|(p2,p2)| - 1)) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (|(p2,p2)| + 1)) * (|(p2,p2)| - 1)) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(((1 / (|(p2,p2)| + 1)) * 2) * p2) + (((1 / (|(p2,p2)| + 1)) * (|(p2,p2)| - 1)) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((1 / (|(p2,p2)| + 1)) * 2) * p2) + (((1 / (|(p2,p2)| + 1)) * (|(p2,p2)| - 1)) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 * (|(p2,p2)| - 1) is V21() real ext-real Element of REAL
(1 * (|(p2,p2)| - 1)) / (|(p2,p2)| + 1) is V21() real ext-real Element of REAL
((1 * (|(p2,p2)| - 1)) / (|(p2,p2)| + 1)) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 * (|(p2,p2)| - 1)) / (|(p2,p2)| + 1)) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((1 - |(U2,M)|) * p2) + (((1 * (|(p2,p2)| - 1)) / (|(p2,p2)| + 1)) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 - |(U2,M)|) * p2) + (((1 * (|(p2,p2)| - 1)) / (|(p2,p2)| + 1)) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(|(U2,M)| * M) - (|(U2,M)| * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(|(U2,M)| * M) - (|(U2,M)| * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U2 - ((|(U2,M)| * M) - (|(U2,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 - ((|(U2,M)| * M) - (|(U2,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U2 - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U2 + (- (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 + (- (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U2 + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 + ((- 1) * (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U2 + (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 + (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U11 is set
m is Relation-like the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of p -valued Function-like quasi_total Element of bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of p:]
m . U11 is set
U2 is set
(n,M,p) . U2 is set
V is set
(n,M,p) . V is set
rng (n,M,p) is Element of bool the carrier of (n,M,(0. (TOP-REAL n)))
U11 is set
dom (n,M,p) is set
V is set
(n,M,p) . U11 is set
(n,M,p) . V is set
U . ((n,M,p) . U11) is set
U . ((n,M,p) . V) is set
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
(n,M) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() continuous 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 Relation-like non empty complex-yielding V117() V118() set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
[: the carrier of (TOP-REAL A), the carrier of R^1:] is Relation-like non empty complex-yielding V117() V118() set
bool [: the carrier of (TOP-REAL A), the carrier of R^1:] is non empty set
V is Relation-like the carrier of (TOP-REAL A) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL A), the carrier of R^1:]
U2 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
p2 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() continuous Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
g4 is non empty TopSpace-like TopStruct
p2 | g4 is Relation-like the carrier of g4 -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of g4, the carrier of R^1:]
the carrier of g4 is non empty set
[: the carrier of g4, the carrier of R^1:] is Relation-like non empty complex-yielding V117() V118() set
bool [: the carrier of g4, the carrier of R^1:] is non empty set
g6 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(p2 | g4) . g6 is V21() real ext-real Element of REAL
|(g6,M)| is V21() real ext-real Element of REAL
1 - |(g6,M)| is V21() real ext-real Element of REAL
[#] g4 is non empty non proper open closed Element of bool the carrier of g4
bool the carrier of g4 is non empty set
(n,M) . g6 is V21() real ext-real Element of REAL
V . g6 is V21() real ext-real Element of REAL
p2 . g6 is V21() real ext-real Element of REAL
p2 | the carrier of g4 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of g4 -defined the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
(p2 | the carrier of g4) . g6 is V21() real ext-real Element of REAL
g6 is Element of the carrier of g4
(p2 | g4) . g6 is V21() real ext-real Element of REAL
g7 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
g8 is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g8 - (0. (TOP-REAL A)) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g8 - (0. (TOP-REAL A)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(g8 - (0. (TOP-REAL A))).| is V21() real ext-real non negative Element of REAL
sqr (g8 - (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g8 - (0. (TOP-REAL A)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g8 - (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
g8 + (- (0. (TOP-REAL A))) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g8 + (- (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(g8 + (- (0. (TOP-REAL A)))).| is V21() real ext-real non negative Element of REAL
sqr (g8 + (- (0. (TOP-REAL A)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g8 + (- (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g8 + (- (0. (TOP-REAL A)))))) is V21() real ext-real Element of REAL
g8 + ((- 1) * (0. (TOP-REAL A))) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g8 + ((- 1) * (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(g8 + ((- 1) * (0. (TOP-REAL A)))).| is V21() real ext-real non negative Element of REAL
sqr (g8 + ((- 1) * (0. (TOP-REAL A)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g8 + ((- 1) * (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g8 + ((- 1) * (0. (TOP-REAL A)))))) is V21() real ext-real Element of REAL
g8 + (0. (TOP-REAL A)) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g8 + (0. (TOP-REAL A)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(g8 + (0. (TOP-REAL A))).| is V21() real ext-real non negative Element of REAL
sqr (g8 + (0. (TOP-REAL A))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (g8 + (0. (TOP-REAL A)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (g8 + (0. (TOP-REAL A))))) is V21() real ext-real Element of REAL
|.g7.| is V21() real ext-real non negative Element of REAL
sqr g7 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr g7) is V21() real ext-real Element of REAL
sqrt (Sum (sqr g7)) is V21() real ext-real Element of REAL
|(g7,g7)| is V21() real ext-real Element of REAL
|(g7,M)| is V21() real ext-real Element of REAL
|(g7,M)| - |(g7,g7)| is V21() real ext-real Element of REAL
M - g7 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
M - g7 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(g7,(M - g7))| is V21() real ext-real Element of REAL
|(M,M)| - |(g7,M)| is V21() real ext-real Element of REAL
|((M - g7),M)| is V21() real ext-real Element of REAL
|((M - g7),(M - g7))| is V21() real ext-real Element of REAL
{} - {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural real finite finite-yielding V36() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding V117() V118() V119() R-orthogonal R-normal R-orthonormal Element of REAL
1 - |(g7,M)| is V21() real ext-real Element of REAL
g6 is Relation-like the carrier of g4 -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of g4, the carrier of R^1:]
[: the carrier of g4, the carrier of (TOP-REAL A):] is Relation-like non empty set
bool [: the carrier of g4, the carrier of (TOP-REAL A):] is non empty set
g7 is Relation-like the carrier of g4 -defined the carrier of (TOP-REAL A) -valued Function-like non empty total quasi_total Element of bool [: the carrier of g4, the carrier of (TOP-REAL A):]
(n,M) | g4 is Relation-like the carrier of g4 -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of g4, the carrier of R^1:]
p1 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((n,M) | g4) . p1 is V21() real ext-real Element of REAL
|(p1,M)| is V21() real ext-real Element of REAL
[#] g4 is non empty non proper open closed Element of bool the carrier of g4
bool the carrier of g4 is non empty set
(n,M) . p1 is V21() real ext-real Element of REAL
(n,M) | the carrier of g4 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of g4 -defined the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
((n,M) | the carrier of g4) . p1 is V21() real ext-real Element of REAL
p1 is Relation-like the carrier of g4 -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of g4, the carrier of R^1:]
g9 is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g10 is Relation-like the carrier of g4 -defined the carrier of (TOP-REAL A) -valued Function-like non empty total quasi_total Element of bool [: the carrier of g4, the carrier of (TOP-REAL A):]
[: the carrier of p, the carrier of (TOP-REAL n):] is Relation-like set
bool [: the carrier of p, the carrier of (TOP-REAL n):] is non empty set
g11 is Relation-like the carrier of p -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total Element of bool [: the carrier of p, the carrier of (TOP-REAL n):]
dom g11 is Element of bool the carrier of p
x is set
(n,M,p) . x is set
g11 . x is Relation-like Function-like set
qx is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
r is Element of the carrier of g4
(p2 | g4) . r is V21() real ext-real Element of REAL
|(qx,M)| is V21() real ext-real Element of REAL
1 - |(qx,M)| is V21() real ext-real Element of REAL
g6 . r is V21() real ext-real Element of REAL
1 / (1 - |(qx,M)|) is V21() real ext-real Element of REAL
((n,M) | g4) . r is V21() real ext-real Element of REAL
g10 . r is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
p1 . r is V21() real ext-real Element of REAL
(p1 . r) * (- M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(p1 . r) * (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(qx,M)| / (1 - |(qx,M)|) is V21() real ext-real Element of REAL
(|(qx,M)| / (1 - |(qx,M)|)) * (- M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(|(qx,M)| / (1 - |(qx,M)|)) * (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
g7 . r is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
(g7 . r) + (g10 . r) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
(g7 . r) + (g10 . r) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (1 - |(qx,M)|)) * qx is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(qx,M)|)) * qx is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 * |(qx,M)| is V21() real ext-real Element of REAL
(1 * |(qx,M)|) / (1 - |(qx,M)|) is V21() real ext-real Element of REAL
((1 * |(qx,M)|) / (1 - |(qx,M)|)) * (- M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 * |(qx,M)|) / (1 - |(qx,M)|)) * (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((1 / (1 - |(qx,M)|)) * qx) + (((1 * |(qx,M)|) / (1 - |(qx,M)|)) * (- M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (1 - |(qx,M)|)) * qx) + (((1 * |(qx,M)|) / (1 - |(qx,M)|)) * (- M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (1 - |(qx,M)|)) * |(qx,M)| is V21() real ext-real Element of REAL
((1 / (1 - |(qx,M)|)) * |(qx,M)|) * (- M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (1 - |(qx,M)|)) * |(qx,M)|) * (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((1 / (1 - |(qx,M)|)) * qx) + (((1 / (1 - |(qx,M)|)) * |(qx,M)|) * (- M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (1 - |(qx,M)|)) * qx) + (((1 / (1 - |(qx,M)|)) * |(qx,M)|) * (- M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(qx,M)| * (- M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(qx,M)| * (- M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (1 - |(qx,M)|)) * (|(qx,M)| * (- M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(qx,M)|)) * (|(qx,M)| * (- M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((1 / (1 - |(qx,M)|)) * qx) + ((1 / (1 - |(qx,M)|)) * (|(qx,M)| * (- M))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (1 - |(qx,M)|)) * qx) + ((1 / (1 - |(qx,M)|)) * (|(qx,M)| * (- M))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
qx + (|(qx,M)| * (- M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
qx + (|(qx,M)| * (- M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (1 - |(qx,M)|)) * (qx + (|(qx,M)| * (- M))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(qx,M)|)) * (qx + (|(qx,M)| * (- M))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(qx,M)| * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
|(qx,M)| * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
- (|(qx,M)| * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
- (|(qx,M)| * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
qx + (- (|(qx,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
qx + (- (|(qx,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (1 - |(qx,M)|)) * (qx + (- (|(qx,M)| * M))) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(qx,M)|)) * (qx + (- (|(qx,M)| * M))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
qx - (|(qx,M)| * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
qx - (|(qx,M)| * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (1 - |(qx,M)|)) * (qx - (|(qx,M)| * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (1 - |(qx,M)|)) * (qx - (|(qx,M)| * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
m is Relation-like the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of p -valued Function-like quasi_total Element of bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of p:]
U11 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
m . U11 is set
2 * U11 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
2 * U11 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(U11,U11)| is V21() real ext-real Element of REAL
|(U11,U11)| - 1 is V21() real ext-real Element of REAL
(|(U11,U11)| - 1) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(|(U11,U11)| - 1) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(2 * U11) + ((|(U11,U11)| - 1) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(2 * U11) + ((|(U11,U11)| - 1) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|(U11,U11)| + 1 is V21() real ext-real Element of REAL
1 / (|(U11,U11)| + 1) is V21() real ext-real Element of REAL
(1 / (|(U11,U11)| + 1)) * ((2 * U11) + ((|(U11,U11)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (|(U11,U11)| + 1)) * ((2 * U11) + ((|(U11,U11)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
[: the carrier of (TOP-REAL A), the carrier of R^1:] is Relation-like non empty complex-yielding V117() V118() set
bool [: the carrier of (TOP-REAL A), the carrier of R^1:] is non empty set
U11 is Relation-like the carrier of (TOP-REAL A) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL A), the carrier of R^1:]
[: the carrier of (TOP-REAL n), the carrier of R^1:] is Relation-like non empty complex-yielding V117() V118() set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
V is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
U2 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
p2 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
g4 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
g5 is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U2 . g5 is V21() real ext-real Element of REAL
U11 . g5 is V21() real ext-real Element of REAL
|.g5.| is V21() real ext-real non negative Element of REAL
sqr g5 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr g5) is V21() real ext-real Element of REAL
sqrt (Sum (sqr g5)) is V21() real ext-real Element of REAL
V . g5 is V21() real ext-real Element of REAL
|.g5.| * |.g5.| is V21() real ext-real non negative Element of REAL
(|.g5.| * |.g5.|) + 1 is non empty V21() real ext-real positive non negative Element of REAL
g5 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
[: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of R^1:] is Relation-like non empty complex-yielding V117() V118() set
bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of R^1:] is non empty set
g5 | (n,M,(0. (TOP-REAL n))) is Relation-like the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of R^1:]
[: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of (TOP-REAL A):] is Relation-like non empty set
bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of (TOP-REAL A):] is non empty set
g6 is Relation-like the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() continuous Element of bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of R^1:]
g7 is Relation-like the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of (TOP-REAL A) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of (TOP-REAL A):]
g8 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
g8 | (n,M,(0. (TOP-REAL n))) is Relation-like the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() Element of bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of R^1:]
p1 is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
g9 is Relation-like the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding V117() V118() continuous Element of bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of R^1:]
g10 is Relation-like the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of (TOP-REAL A) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of (TOP-REAL A):]
g11 is Relation-like the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of (TOP-REAL A) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (n,M,(0. (TOP-REAL n))), the carrier of (TOP-REAL A):]
dom m is Element of bool the carrier of (n,M,(0. (TOP-REAL n)))
dom g11 is non empty Element of bool the carrier of (n,M,(0. (TOP-REAL n)))
x is set
m . x is set
g11 . x is Relation-like Function-like set
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
qx is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
U11 . qx is V21() real ext-real Element of REAL
|.qx.| is V21() real ext-real non negative Element of REAL
sqr qx is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr qx) is V21() real ext-real Element of REAL
sqrt (Sum (sqr qx)) is V21() real ext-real Element of REAL
V . qx is V21() real ext-real Element of REAL
|.qx.| * |.qx.| is V21() real ext-real non negative Element of REAL
|.qx.| ^2 is V21() real ext-real Element of REAL
|(qx,qx)| is V21() real ext-real Element of REAL
U2 . qx is V21() real ext-real Element of REAL
|(qx,qx)| + 1 is V21() real ext-real Element of REAL
g4 . qx is V21() real ext-real Element of REAL
g6 . qx is V21() real ext-real Element of REAL
g5 | the carrier of (n,M,(0. (TOP-REAL n))) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
(g5 | the carrier of (n,M,(0. (TOP-REAL n)))) . qx is V21() real ext-real Element of REAL
g5 . qx is V21() real ext-real Element of REAL
2 / (|(qx,qx)| + 1) is V21() real ext-real Element of REAL
p2 . qx is V21() real ext-real Element of REAL
|(qx,qx)| + (- 1) is V21() real ext-real Element of REAL
g9 . qx is V21() real ext-real Element of REAL
g8 | the carrier of (n,M,(0. (TOP-REAL n))) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (n,M,(0. (TOP-REAL n))) -defined the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like complex-yielding V117() V118() Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
(g8 | the carrier of (n,M,(0. (TOP-REAL n)))) . qx is V21() real ext-real Element of REAL
g8 . qx is V21() real ext-real Element of REAL
|(qx,qx)| - 1 is V21() real ext-real Element of REAL
(|(qx,qx)| - 1) / (|(qx,qx)| + 1) is V21() real ext-real Element of REAL
r is Element of the carrier of (n,M,(0. (TOP-REAL n)))
g7 . r is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
(2 / (|(qx,qx)| + 1)) * qx is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(2 / (|(qx,qx)| + 1)) * qx is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
g10 . r is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
(g7 . r) + (g10 . r) is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL A)
(g7 . r) + (g10 . r) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 * 2 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
(1 * 2) / (|(qx,qx)| + 1) is V21() real ext-real Element of REAL
((1 * 2) / (|(qx,qx)| + 1)) * qx is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 * 2) / (|(qx,qx)| + 1)) * qx is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 * (|(qx,qx)| - 1) is V21() real ext-real Element of REAL
(1 * (|(qx,qx)| - 1)) / (|(qx,qx)| + 1) is V21() real ext-real Element of REAL
((1 * (|(qx,qx)| - 1)) / (|(qx,qx)| + 1)) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 * (|(qx,qx)| - 1)) / (|(qx,qx)| + 1)) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(((1 * 2) / (|(qx,qx)| + 1)) * qx) + (((1 * (|(qx,qx)| - 1)) / (|(qx,qx)| + 1)) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((1 * 2) / (|(qx,qx)| + 1)) * qx) + (((1 * (|(qx,qx)| - 1)) / (|(qx,qx)| + 1)) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 / (|(qx,qx)| + 1) is V21() real ext-real Element of REAL
(1 / (|(qx,qx)| + 1)) * (|(qx,qx)| - 1) is V21() real ext-real Element of REAL
((1 / (|(qx,qx)| + 1)) * (|(qx,qx)| - 1)) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (|(qx,qx)| + 1)) * (|(qx,qx)| - 1)) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(((1 * 2) / (|(qx,qx)| + 1)) * qx) + (((1 / (|(qx,qx)| + 1)) * (|(qx,qx)| - 1)) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((1 * 2) / (|(qx,qx)| + 1)) * qx) + (((1 / (|(qx,qx)| + 1)) * (|(qx,qx)| - 1)) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (|(qx,qx)| + 1)) * 2 is V21() real ext-real Element of REAL
((1 / (|(qx,qx)| + 1)) * 2) * qx is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (|(qx,qx)| + 1)) * 2) * qx is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(((1 / (|(qx,qx)| + 1)) * 2) * qx) + (((1 / (|(qx,qx)| + 1)) * (|(qx,qx)| - 1)) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((1 / (|(qx,qx)| + 1)) * 2) * qx) + (((1 / (|(qx,qx)| + 1)) * (|(qx,qx)| - 1)) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(|(qx,qx)| - 1) * M is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(|(qx,qx)| - 1) * M is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (|(qx,qx)| + 1)) * ((|(qx,qx)| - 1) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (|(qx,qx)| + 1)) * ((|(qx,qx)| - 1) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(((1 / (|(qx,qx)| + 1)) * 2) * qx) + ((1 / (|(qx,qx)| + 1)) * ((|(qx,qx)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(((1 / (|(qx,qx)| + 1)) * 2) * qx) + ((1 / (|(qx,qx)| + 1)) * ((|(qx,qx)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
2 * qx is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
2 * qx is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (|(qx,qx)| + 1)) * (2 * qx) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (|(qx,qx)| + 1)) * (2 * qx) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
((1 / (|(qx,qx)| + 1)) * (2 * qx)) + ((1 / (|(qx,qx)| + 1)) * ((|(qx,qx)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
((1 / (|(qx,qx)| + 1)) * (2 * qx)) + ((1 / (|(qx,qx)| + 1)) * ((|(qx,qx)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(2 * qx) + ((|(qx,qx)| - 1) * M) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(2 * qx) + ((|(qx,qx)| - 1) * M) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 / (|(qx,qx)| + 1)) * ((2 * qx) + ((|(qx,qx)| - 1) * M)) is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n)
(1 / (|(qx,qx)| + 1)) * ((2 * qx) + ((|(qx,qx)| - 1) * M)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U11 is Relation-like [#] (n,M,(0. (TOP-REAL n))) -valued set
S /" is Relation-like [#] (n,M,(0. (TOP-REAL n))) -defined [#] p -valued Function-like quasi_total Element of bool [:([#] (n,M,(0. (TOP-REAL n)))),([#] p):]
S " is Relation-like Function-like set
n is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
(n) is non empty TopSpace-like TopStruct
n + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
Tunit_circle (n + 1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL (n + 1)
TOP-REAL (n + 1) is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n + 1 -locally_euclidean n + 1 -manifold manifold-like RLTopStruct
0. (TOP-REAL (n + 1)) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like V51( TOP-REAL (n + 1)) complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
the carrier of (TOP-REAL (n + 1)) is functional non empty set
the ZeroF of (TOP-REAL (n + 1)) is Relation-like NAT -defined Function-like finite n + 1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL (n + 1))
Sphere ((0. (TOP-REAL (n + 1))),1) is functional non empty closed V199( TOP-REAL (n + 1)) Element of bool the carrier of (TOP-REAL (n + 1))
bool the carrier of (TOP-REAL (n + 1)) is non empty set
(TOP-REAL (n + 1)) | (Sphere ((0. (TOP-REAL (n + 1))),1)) is non empty strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL (n + 1)
Tcircle ((0. (TOP-REAL (n + 1))),1) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL (n + 1)
the carrier of (n) is non empty set
TOP-REAL n is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n -locally_euclidean n -manifold manifold-like RLTopStruct
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
p is Element of the carrier of (n)
the carrier of (Tunit_circle (n + 1)) is non empty set
[#] (Tunit_circle (n + 1)) is non empty non proper open closed Element of bool the carrier of (Tunit_circle (n + 1))
bool the carrier of (Tunit_circle (n + 1)) is non empty set
[#] (TOP-REAL (n + 1)) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL (n + 1))
the carrier of (TOP-REAL (n + 1)) is functional non empty set
bool the carrier of (TOP-REAL (n + 1)) is non empty set
n1 is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative Element of NAT
TOP-REAL n1 is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional n1 -locally_euclidean n1 -manifold manifold-like RLTopStruct
the carrier of (TOP-REAL n1) is functional non empty set
p1 is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
|.p1.| is V21() real ext-real non negative Element of REAL
sqr p1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr p1) is V21() real ext-real Element of REAL
sqrt (Sum (sqr p1)) is V21() real ext-real Element of REAL
- p1 is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
- p1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.(- p1).| is V21() real ext-real non negative Element of REAL
sqr (- p1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (- p1)) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (- p1))) is V21() real ext-real Element of REAL
|((- p1),(- p1))| is V21() real ext-real Element of REAL
sqrt |((- p1),(- p1))| is V21() real ext-real Element of REAL
|(p1,p1)| is V21() real ext-real Element of REAL
sqrt |(p1,p1)| is V21() real ext-real Element of REAL
0. (TOP-REAL n1) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like V51( TOP-REAL n1) complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
the ZeroF of (TOP-REAL n1) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(- p1) + (0. (TOP-REAL n1)) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(- p1) + (0. (TOP-REAL n1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.((- p1) + (0. (TOP-REAL n1))).| is V21() real ext-real non negative Element of REAL
sqr ((- p1) + (0. (TOP-REAL n1))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr ((- p1) + (0. (TOP-REAL n1)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr ((- p1) + (0. (TOP-REAL n1))))) is V21() real ext-real Element of REAL
- 1 is non empty V21() real ext-real non positive negative Element of REAL
(- 1) * (0. (TOP-REAL n1)) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(- 1) * (0. (TOP-REAL n1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(- p1) + ((- 1) * (0. (TOP-REAL n1))) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(- p1) + ((- 1) * (0. (TOP-REAL n1))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.((- p1) + ((- 1) * (0. (TOP-REAL n1)))).| is V21() real ext-real non negative Element of REAL
sqr ((- p1) + ((- 1) * (0. (TOP-REAL n1)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr ((- p1) + ((- 1) * (0. (TOP-REAL n1))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr ((- p1) + ((- 1) * (0. (TOP-REAL n1)))))) is V21() real ext-real Element of REAL
- (0. (TOP-REAL n1)) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
- (0. (TOP-REAL n1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(- p1) + (- (0. (TOP-REAL n1))) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(- p1) + (- (0. (TOP-REAL n1))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.((- p1) + (- (0. (TOP-REAL n1)))).| is V21() real ext-real non negative Element of REAL
sqr ((- p1) + (- (0. (TOP-REAL n1)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr ((- p1) + (- (0. (TOP-REAL n1))))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr ((- p1) + (- (0. (TOP-REAL n1)))))) is V21() real ext-real Element of REAL
(- p1) - (0. (TOP-REAL n1)) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(- p1) - (0. (TOP-REAL n1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
|.((- p1) - (0. (TOP-REAL n1))).| is V21() real ext-real non negative Element of REAL
sqr ((- p1) - (0. (TOP-REAL n1))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr ((- p1) - (0. (TOP-REAL n1)))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr ((- p1) - (0. (TOP-REAL n1))))) is V21() real ext-real Element of REAL
Sphere ((0. (TOP-REAL n1)),1) is functional closed V199( TOP-REAL n1) Element of bool the carrier of (TOP-REAL n1)
bool the carrier of (TOP-REAL n1) is non empty set
(TOP-REAL n1) | (Sphere ((0. (TOP-REAL n1)),1)) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL n1
[#] ((TOP-REAL n1) | (Sphere ((0. (TOP-REAL n1)),1))) is non proper open closed Element of bool the carrier of ((TOP-REAL n1) | (Sphere ((0. (TOP-REAL n1)),1)))
the carrier of ((TOP-REAL n1) | (Sphere ((0. (TOP-REAL n1)),1))) is set
bool the carrier of ((TOP-REAL n1) | (Sphere ((0. (TOP-REAL n1)),1))) is non empty set
Tcircle ((0. (TOP-REAL n1)),1) is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL n1
the carrier of (Tcircle ((0. (TOP-REAL n1)),1)) is set
Tunit_circle n1 is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL n1
the carrier of (Tunit_circle n1) is set
bool the carrier of (n) is non empty set
{(- p1)} is functional non empty finite V36() set
[#] (n) is non empty non proper open closed Element of bool the carrier of (n)
A is Element of bool the carrier of (n)
([#] (n)) \ A is Element of bool the carrier of (n)
|.(0. (TOP-REAL n1)).| is V21() real ext-real non negative Element of REAL
sqr (0. (TOP-REAL n1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
Sum (sqr (0. (TOP-REAL n1))) is V21() real ext-real Element of REAL
sqrt (Sum (sqr (0. (TOP-REAL n1)))) is V21() real ext-real Element of REAL
1 + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural real ext-real positive non negative Element of NAT
(1 + 1) * (- p1) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(1 + 1) * (- p1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
1 * (- p1) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
1 * (- p1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 * (- p1)) + (1 * (- p1)) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(1 * (- p1)) + (1 * (- p1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(1 * (- p1)) + (- p1) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(1 * (- p1)) + (- p1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(- p1) + (- p1) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(- p1) + (- p1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
- (- p1) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
- (- p1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
(- p1) + (- (- p1)) is Relation-like NAT -defined Function-like finite n1 -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL n1)
(- p1) + (- (- p1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-yielding V117() V118() FinSequence of REAL
U1 is Element of bool the carrier of (n)
U is a_neighborhood of p
m is V21() epsilon-transitive epsilon-connected ordinal natural real ext-real non negative set
TOP-REAL m is non empty V70() V138() V139() TopSpace-like T_0 T_1 T_2 Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict add-continuous Mult-continuous V234() second-countable finite-dimensional m -locally_euclidean m -manifold manifold-like RLTopStruct
0. (TOP-REAL m) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like V51( TOP-REAL m) complex-yielding V117() V118() Element of the carrier of (TOP-REAL m)
the carrier of (TOP-REAL m) is functional non empty set
the ZeroF of (TOP-REAL m) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL m)
Tcircle ((0. (TOP-REAL m)),1) is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL m
Sphere ((0. (TOP-REAL m)),1) is functional closed Element of bool the carrier of (TOP-REAL m)
bool the carrier of (TOP-REAL m) is non empty set
(TOP-REAL m) | (Sphere ((0. (TOP-REAL m)),1)) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL m
S is functional Element of bool the carrier of (TOP-REAL m)
(TOP-REAL m) | S is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL m
the carrier of ((TOP-REAL m) | S) is set
bool the carrier of ((TOP-REAL m) | S) is non empty set
Tunit_circle m is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL m
[#] (Tunit_circle m) is non proper open closed Element of bool the carrier of (Tunit_circle m)
the carrier of (Tunit_circle m) is set
bool the carrier of (Tunit_circle m) is non empty set
U11 is Element of bool the carrier of ((TOP-REAL m) | S)
(n) | U is strict TopSpace-like V234() second-countable SubSpace of (n)
V is functional non empty Element of bool the carrier of (TOP-REAL m)
(TOP-REAL m) | V is non empty strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL m
p2 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL m)
(m,p2,(0. (TOP-REAL m))) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL m
(m,p2,(0. (TOP-REAL m))) is functional Element of bool the carrier of (TOP-REAL m)
{ b₁ where b₁ is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-yielding V117() V118() Element of the carrier of (TOP-REAL m) : |(p2,(b₁ - (0. (TOP-REAL m))))| = {} } is set
(TOP-REAL m) | (m,p2,(0. (TOP-REAL m))) is strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL m
(TOP-REAL n) | ([#] (TOP-REAL n)) is non empty strict TopSpace-like T_0 T_1 T_2 V234() second-countable SubSpace of TOP-REAL n
the topology of (TOP-REAL n) is non empty open 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 V234() second-countable TopStruct
U2 is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL m
[#] U2 is non empty non proper open closed Element of bool the carrier of U2
the carrier of U2 is non empty set
bool the carrier of U2 is non empty set
the carrier of (Tcircle ((0. (TOP-REAL m)),1)) is set
{p2} is functional non empty finite V36() set
the carrier of (Tcircle ((0. (TOP-REAL m)),1)) \ {p2} is Element of bool the carrier of (Tcircle ((0. (TOP-REAL m)),1))
bool the carrier of (Tcircle ((0. (TOP-REAL m)),1)) is non empty set
the carrier of ((TOP-REAL m) | (Sphere ((0. (TOP-REAL m)),1))) is set
[#] ((TOP-REAL m) | (Sphere ((0. (TOP-REAL m)),1))) is non proper open closed Element of bool the carrier of ((TOP-REAL m) | (Sphere ((0. (TOP-REAL m)),1)))
bool the carrier of ((TOP-REAL m) | (Sphere ((0. (TOP-REAL m)),1))) is non empty set
([#] ((TOP-REAL m) | (Sphere ((0. (TOP-REAL m)),1)))) \ {p2} is Element of bool the carrier of ((TOP-REAL m) | (Sphere ((0. (TOP-REAL m)),1)))
(Sphere ((0. (TOP-REAL m)),1)) \ {p2} is functional Element of bool the carrier of (TOP-REAL m)
(m,p2,U2) is Relation-like the carrier of U2 -defined the carrier of (m,p2,(0. (TOP-REAL m))) -valued Function-like non empty total quasi_total Element of bool [: the carrier of U2, the carrier of (m,p2,(0. (TOP-REAL m))):]
the carrier of (m,p2,(0. (TOP-REAL m))) is non empty set
[: the carrier of U2, the carrier of (m,p2,(0. (TOP-REAL m))):] is Relation-like non empty set
bool [: the carrier of U2, the carrier of (m,p2,(0. (TOP-REAL m))):] is non empty set