:: MFOLD_1 semantic presentation

REAL is complex-membered ext-real-membered real-membered V140() non bounded_below non bounded_above V317() set
NAT is ordinal V35() cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below Element of bool REAL
bool REAL is non empty set
omega is ordinal V35() cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below set
bool omega is non empty V35() set
bool NAT is non empty V35() set
1 is non empty ordinal natural V11() real ext-real positive non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of NAT
[:1,1:] is non empty Relation-like RAT -valued INT -valued V124() V125() V126() V127() set
RAT is complex-membered ext-real-membered real-membered rational-membered V140() set
INT is complex-membered ext-real-membered real-membered rational-membered integer-membered V140() set
bool [:1,1:] is non empty set
[:[:1,1:],1:] is non empty Relation-like RAT -valued INT -valued V124() V125() V126() V127() set
bool [:[:1,1:],1:] is non empty set
[:[:1,1:],REAL:] is Relation-like V124() V125() V126() set
bool [:[:1,1:],REAL:] is non empty set
[:REAL,REAL:] is Relation-like V124() V125() V126() set
[:[:REAL,REAL:],REAL:] is Relation-like V124() V125() V126() set
bool [:[:REAL,REAL:],REAL:] is non empty set
2 is non empty ordinal natural V11() real ext-real positive non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of NAT
[:2,2:] is non empty Relation-like RAT -valued INT -valued V124() V125() V126() V127() set
[:[:2,2:],REAL:] is Relation-like V124() V125() V126() set
bool [:[:2,2:],REAL:] is non empty set
COMPLEX is complex-membered V140() set
bool [:REAL,REAL:] is non empty set
TOP-REAL 2 is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable L15()
the carrier of (TOP-REAL 2) is non empty functional set
K405() is real-membered TopStruct
the carrier of K405() is complex-membered ext-real-membered real-membered set
RealSpace is strict real-membered MetrStruct
R^1 is non empty strict TopSpace-like real-membered TopStruct
K412() is TopSpace-like real-membered SubSpace of R^1
K390(K412(),K412()) is strict TopSpace-like TopStruct
the carrier of K390(K412(),K412()) is set
K414() is non empty strict TopSpace-like real-membered V191() SubSpace of R^1
the carrier of K414() is non empty complex-membered ext-real-membered real-membered set
bool the carrier of K414() is non empty set
bool (bool the carrier of K414()) is non empty set
K423(2) is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL 2
the carrier of K423(2) is set
[: the carrier of K414(), the carrier of K423(2):] is Relation-like set
bool [: the carrier of K414(), the carrier of K423(2):] is non empty set
K427() is Relation-like the carrier of K414() -defined the carrier of K423(2) -valued Function-like quasi_total Element of bool [: the carrier of K414(), the carrier of K423(2):]
K424() is Element of the carrier of K423(2)
K426(K424()) is strict TopSpace-like T_0 T_1 T_2 SubSpace of K423(2)
the carrier of K426(K424()) is set
0 is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V33() V34() V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of NAT
{} is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() set
K328(0,1) is complex-membered ext-real-membered real-membered non left_end non right_end V317() Element of bool REAL
K420(K328(0,1)) is complex-membered ext-real-membered real-membered Element of bool the carrier of K414()
K414() | K420(K328(0,1)) is strict TopSpace-like real-membered SubSpace of K414()
the carrier of (K414() | K420(K328(0,1))) is complex-membered ext-real-membered real-membered set
[: the carrier of K426(K424()), the carrier of (K414() | K420(K328(0,1))):] is Relation-like V124() V125() V126() set
bool [: the carrier of K426(K424()), the carrier of (K414() | K420(K328(0,1))):] is non empty set
K425() is Element of the carrier of K423(2)
K426(K425()) is strict TopSpace-like T_0 T_1 T_2 SubSpace of K423(2)
the carrier of K426(K425()) is set
1 / 2 is non empty V11() real ext-real positive non negative Element of REAL
2 " is non empty V11() real ext-real positive non negative set
1 * (2 ") is non empty V11() real ext-real positive non negative set
3 is non empty ordinal natural V11() real ext-real positive non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of NAT
3 / 2 is non empty V11() real ext-real positive non negative Element of REAL
3 * (2 ") is non empty V11() real ext-real positive non negative set
K328((1 / 2),(3 / 2)) is complex-membered ext-real-membered real-membered non left_end non right_end V317() Element of bool REAL
K420(K328((1 / 2),(3 / 2))) is complex-membered ext-real-membered real-membered Element of bool the carrier of K414()
K414() | K420(K328((1 / 2),(3 / 2))) is strict TopSpace-like real-membered SubSpace of K414()
the carrier of (K414() | K420(K328((1 / 2),(3 / 2)))) is complex-membered ext-real-membered real-membered set
[: the carrier of K426(K425()), the carrier of (K414() | K420(K328((1 / 2),(3 / 2)))):] is Relation-like V124() V125() V126() set
bool [: the carrier of K426(K425()), the carrier of (K414() | K420(K328((1 / 2),(3 / 2)))):] is non empty set
[:COMPLEX,COMPLEX:] is Relation-like V124() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like V124() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:RAT,RAT:] is Relation-like RAT -valued V124() V125() V126() set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is Relation-like RAT -valued V124() V125() V126() set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is Relation-like RAT -valued INT -valued V124() V125() V126() set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is Relation-like RAT -valued INT -valued V124() V125() V126() set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is Relation-like RAT -valued INT -valued V124() V125() V126() V127() set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued V124() V125() V126() V127() set
bool [:[:NAT,NAT:],NAT:] is non empty set
[:COMPLEX,REAL:] is Relation-like V124() V125() V126() set
bool [:COMPLEX,REAL:] is non empty set
REAL 0 is non empty functional V45() M11( REAL )
TOP-REAL 0 is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable L15()
0. (TOP-REAL 0) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal 0 -element V43() V44() V45() V54( TOP-REAL 0) V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL 0)
the carrier of (TOP-REAL 0) is non empty functional set
the ZeroF of (TOP-REAL 0) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal 0 -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL 0)
{(0. (TOP-REAL 0))} is non empty trivial functional 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
the carrier of R^1 is non empty complex-membered ext-real-membered real-membered set
sqrt 1 is V11() real ext-real Element of REAL
- 1 is non empty V11() real ext-real non positive negative set
bool the carrier of R^1 is non empty set
T is set
c₂ is set
[T,c₂] is set
{T,c₂} is non empty set
{T} is non empty trivial 1 -element set
{{T,c₂},{T}} is non empty set
{[T,c₂]} is non empty trivial Relation-like Function-like constant 1 -element set
X is set
dom {[T,c₂]} is non empty trivial 1 -element set
X1 is set
{[T,c₂]} . X is set
{[T,c₂]} . X1 is set
T is non empty TopSpace-like TopStruct
[#] T is non empty non proper open closed dense non boundary Element of bool the carrier of T
the carrier of T is non empty set
bool the carrier of T is non empty set
T | ([#] T) is non empty strict TopSpace-like SubSpace of T
id T is non empty Relation-like the carrier of T -defined the carrier of T -valued Function-like V26( the carrier of T) quasi_total continuous Element of bool [: the carrier of T, the carrier of T:]
[: the carrier of T, the carrier of T:] is non empty Relation-like set
bool [: the carrier of T, the carrier of T:] is non empty set
K66( the carrier of T) is non empty Relation-like the carrier of T -defined the carrier of T -valued Function-like one-to-one V26( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of T:]
dom (id T) is non empty Element of bool the carrier of T
[#] (T | ([#] T)) is non empty non proper open closed dense non boundary Element of bool the carrier of (T | ([#] T))
the carrier of (T | ([#] T)) is non empty set
bool the carrier of (T | ([#] T)) is non empty set
the topology of T is non empty open Element of bool (bool the carrier of T)
bool (bool the carrier of T) is non empty set
TopStruct(# the carrier of T, the topology of T #) is non empty strict TopSpace-like TopStruct
the carrier of TopStruct(# the carrier of T, the topology of T #) is non empty set
rng (id T) is non empty Element of bool the carrier of T
[: the carrier of T, the carrier of (T | ([#] T)):] is non empty Relation-like set
bool [: the carrier of T, the carrier of (T | ([#] T)):] is non empty set
M is non empty Relation-like the carrier of T -defined the carrier of (T | ([#] T)) -valued Function-like V26( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of (T | ([#] T)):]
M " is non empty Relation-like the carrier of (T | ([#] T)) -defined the carrier of T -valued Function-like V26( the carrier of (T | ([#] T))) quasi_total Element of bool [: the carrier of (T | ([#] T)), the carrier of T:]
[: the carrier of (T | ([#] T)), the carrier of T:] is non empty Relation-like set
bool [: the carrier of (T | ([#] T)), the carrier of T:] is non empty set
X is Element of bool the carrier of T
(M ") " X is Element of bool the carrier of (T | ([#] T))
([#] T) \ X is Element of bool the carrier of T
X1 is set
M .: X is Element of bool the carrier of (T | ([#] T))
p is set
[p,X1] is set
{p,X1} is non empty set
{p} is non empty trivial 1 -element set
{{p,X1},{p}} is non empty set
[X1,X1] is set
{X1,X1} is non empty set
{X1} is non empty trivial 1 -element set
{{X1,X1},{X1}} is non empty set
([#] (T | ([#] T))) \ ((M ") " X) is Element of bool the carrier of (T | ([#] T))
([#] T) /\ ([#] T) is open closed Element of bool the carrier of T
(([#] T) /\ ([#] T)) \ X is Element of bool the carrier of T
(([#] T) \ X) /\ ([#] T) is Element of bool the carrier of T
the topology of (T | ([#] T)) is non empty open Element of bool (bool the carrier of (T | ([#] T)))
bool (bool the carrier of (T | ([#] T))) is non empty set
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
c₂ is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the carrier of c₂ is non empty set
[: the carrier of c₂, the carrier of R^1:] is non empty Relation-like V124() V125() V126() set
bool [: the carrier of c₂, the carrier of R^1:] is non empty set
[: the carrier of c₂, the carrier of (TOP-REAL T):] is non empty Relation-like set
bool [: the carrier of c₂, the carrier of (TOP-REAL T):] is non empty set
M is non empty Relation-like the carrier of c₂ -defined the carrier of R^1 -valued Function-like V26( the carrier of c₂) quasi_total V124() V125() V126() Element of bool [: the carrier of c₂, the carrier of R^1:]
X is Element of the carrier of c₂
M . X is V11() real ext-real set
M . X is V11() real ext-real Element of the carrier of R^1
[#] c₂ is non empty non proper open closed dense non boundary Element of bool the carrier of c₂
bool the carrier of c₂ is non empty set
[#] (TOP-REAL T) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
p is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
X1 is V11() real ext-real Element of REAL
X1 * p is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
U1 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
S1 is V11() real ext-real set
S1 * U1 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
X is non empty Relation-like the carrier of c₂ -defined the carrier of (TOP-REAL T) -valued Function-like V26( the carrier of c₂) quasi_total Element of bool [: the carrier of c₂, the carrier of (TOP-REAL T):]
X is non empty Relation-like the carrier of c₂ -defined the carrier of (TOP-REAL T) -valued Function-like V26( the carrier of c₂) quasi_total Element of bool [: the carrier of c₂, the carrier of (TOP-REAL T):]
bool the carrier of (TOP-REAL T) is non empty set
bool the carrier of c₂ is non empty set
X1 is Element of the carrier of c₂
X . X1 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
p is functional Element of bool the carrier of (TOP-REAL T)
Euclid T is non empty strict Reflexive discerning V83() triangle MetrStruct
the carrier of (Euclid T) is non empty set
[#] c₂ is non empty non proper open closed dense non boundary Element of bool the carrier of c₂
[#] (TOP-REAL T) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL T)
p1 is ordinal natural V11() real ext-real non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of NAT
TOP-REAL p1 is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable L15()
the carrier of (TOP-REAL p1) is non empty functional set
M . X1 is V11() real ext-real Element of the carrier of R^1
Int p is functional open Element of bool the carrier of (TOP-REAL T)
U1 is Element of the carrier of (Euclid T)
S2 is V11() real ext-real set
Ball (U1,S2) is Element of bool the carrier of (Euclid T)
bool the carrier of (Euclid T) is non empty set
U2 is V11() real ext-real set
S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
S2 / 2 is V11() real ext-real Element of REAL
S2 * (2 ") is V11() real ext-real set
Ball (S1,(S2 / 2)) is functional open V162( TOP-REAL p1) Element of bool the carrier of (TOP-REAL p1)
bool the carrier of (TOP-REAL p1) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1) : not S2 / 2 <= |.(b₁ - S1).| } is set
(Ball (S1,(S2 / 2))) /\ ([#] c₂) is Element of bool the carrier of c₂
the topology of (TOP-REAL p1) is non empty open Element of bool (bool the carrier of (TOP-REAL p1))
bool (bool the carrier of (TOP-REAL p1)) is non empty set
X2 is Element of bool the carrier of c₂
the topology of c₂ is non empty open Element of bool (bool the carrier of c₂)
bool (bool the carrier of c₂) is non empty set
{ |.b₁.| where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : b₁ in X2 } is set
|.S1.| is V11() real ext-real non negative Element of REAL
f is set
U is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.U.| is V11() real ext-real non negative Element of REAL
f is non empty complex-membered ext-real-membered real-membered set
|.S1.| + (S2 / 2) is V11() real ext-real Element of REAL
U is ext-real set
U3 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.U3.| is V11() real ext-real non negative Element of REAL
U3 - S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
- S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),U3,(- S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
the U7 of (TOP-REAL p1) is non empty Relation-like [: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] -defined the carrier of (TOP-REAL p1) -valued Function-like V26([: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):]
[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty set
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),U3,(- S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(U3 - S1).| is V11() real ext-real non negative Element of REAL
|.(- S1).| is V11() real ext-real non negative Element of REAL
|.U3.| - |.(- S1).| is V11() real ext-real Element of REAL
- |.(- S1).| is V11() real ext-real non positive set
|.U3.| + (- |.(- S1).|) is V11() real ext-real set
U3 + (- S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(U3 + (- S1)).| is V11() real ext-real non negative Element of REAL
|.U3.| - |.S1.| is V11() real ext-real Element of REAL
- |.S1.| is V11() real ext-real non positive set
|.U3.| + (- |.S1.|) is V11() real ext-real set
(|.U3.| - |.S1.|) + |.S1.| is V11() real ext-real Element of REAL
(S2 / 2) + |.S1.| is V11() real ext-real Element of REAL
sup f is ext-real set
U is V11() real ext-real set
the topology of R^1 is non empty open Element of bool (bool the carrier of R^1)
bool (bool the carrier of R^1) is non empty set
M2 is open complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
T2 is Element of bool the carrier of c₂
M .: T2 is complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
T2 /\ X2 is Element of bool the carrier of c₂
f2 is Element of bool the carrier of c₂
X .: f2 is functional Element of bool the carrier of (TOP-REAL T)
S is set
dom X is non empty Element of bool the carrier of c₂
Q is set
X . Q is Relation-like Function-like set
S is Element of the carrier of c₂
M . S is V11() real ext-real Element of the carrier of R^1
U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U5 is V11() real ext-real set
U5 * U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
y is Element of the carrier of (Euclid T)
q is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U2 * S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U5 - U2 is V11() real ext-real set
- U2 is V11() real ext-real set
U5 + (- U2) is V11() real ext-real set
abs (U5 - U2) is V11() real ext-real non negative Element of REAL
|.U4.| is V11() real ext-real non negative Element of REAL
(abs (U5 - U2)) * |.U4.| is V11() real ext-real non negative Element of REAL
r2 is V11() real ext-real Element of REAL
abs r2 is V11() real ext-real non negative Element of REAL
(abs r2) * |.U4.| is V11() real ext-real non negative Element of REAL
(U5 - U2) * U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.((U5 - U2) * U4).| is V11() real ext-real non negative Element of REAL
0. (TOP-REAL p1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V54( TOP-REAL p1) V124() V125() V126() Element of the carrier of (TOP-REAL p1)
the ZeroF of (TOP-REAL p1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
gqq is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
- q is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
gqq + (- q) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
the U7 of (TOP-REAL p1) is non empty Relation-like [: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] -defined the carrier of (TOP-REAL p1) -valued Function-like V26([: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):]
[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty set
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),gqq,(- q)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(gqq + (- q)).| is V11() real ext-real non negative Element of REAL
|.gqq.| is V11() real ext-real non negative Element of REAL
|.(- q).| is V11() real ext-real non negative Element of REAL
|.gqq.| + |.(- q).| is V11() real ext-real non negative Element of REAL
|.(0. (TOP-REAL p1)).| is V11() real ext-real non negative Element of REAL
|.gqq.| + |.(0. (TOP-REAL p1)).| is V11() real ext-real non negative Element of REAL
|.gqq.| + {} is V11() real ext-real non negative Element of REAL
gqq - q is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),gqq,(- q)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(gqq - q).| is V11() real ext-real non negative Element of REAL
Ball (q,S2) is functional open V162( TOP-REAL p1) Element of bool the carrier of (TOP-REAL p1)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1) : not S2 <= |.(b₁ - q).| } is set
S2 / U is V11() real ext-real Element of COMPLEX
U " is V11() real ext-real set
S2 * (U ") is V11() real ext-real set
U2 - (S2 / U) is V11() real ext-real set
- (S2 / U) is V11() real ext-real set
U2 + (- (S2 / U)) is V11() real ext-real set
U2 + (S2 / U) is V11() real ext-real set
].(U2 - (S2 / U)),(U2 + (S2 / U)).[ is non left_end non right_end V317() set
(S2 / U) + U2 is V11() real ext-real set
{} + U2 is V11() real ext-real set
- (S2 / U) is V11() real ext-real Element of COMPLEX
(- (S2 / U)) + U2 is V11() real ext-real set
M2 is open complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
T2 is Element of bool the carrier of c₂
M .: T2 is complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
T2 /\ X2 is Element of bool the carrier of c₂
f2 is Element of bool the carrier of c₂
X .: f2 is functional Element of bool the carrier of (TOP-REAL T)
S is set
dom X is non empty Element of bool the carrier of c₂
Q is set
X . Q is Relation-like Function-like set
S is Element of the carrier of c₂
M . S is V11() real ext-real Element of the carrier of R^1
U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U5 is V11() real ext-real set
U5 * U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
y is Element of the carrier of (Euclid T)
q is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U2 * S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
abs U5 is V11() real ext-real non negative Element of REAL
|.U4.| is V11() real ext-real non negative Element of REAL
(abs U5) * |.U4.| is V11() real ext-real non negative Element of REAL
r2 is V11() real ext-real Element of REAL
abs r2 is V11() real ext-real non negative Element of REAL
(abs r2) * |.U4.| is V11() real ext-real non negative Element of REAL
|.(U5 * U4).| is V11() real ext-real non negative Element of REAL
dom M is non empty Element of bool the carrier of c₂
U5 - U2 is V11() real ext-real set
- U2 is V11() real ext-real set
U5 + (- U2) is V11() real ext-real set
abs (U5 - U2) is V11() real ext-real non negative Element of REAL
(S2 / U) * U is V11() real ext-real set
(abs U5) * U is V11() real ext-real Element of REAL
U / U is V11() real ext-real Element of COMPLEX
U * (U ") is V11() real ext-real set
S2 / (U / U) is V11() real ext-real Element of COMPLEX
(U / U) " is V11() real ext-real set
S2 * ((U / U) ") is V11() real ext-real set
S2 / 1 is V11() real ext-real Element of REAL
1 " is non empty V11() real ext-real positive non negative set
S2 * (1 ") is V11() real ext-real set
|.U4.| * (abs U5) is V11() real ext-real non negative Element of REAL
U * (abs U5) is V11() real ext-real Element of REAL
0. (TOP-REAL p1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V54( TOP-REAL p1) V124() V125() V126() Element of the carrier of (TOP-REAL p1)
the ZeroF of (TOP-REAL p1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
gqq is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.gqq.| is V11() real ext-real non negative Element of REAL
- q is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
gqq + (- q) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
the U7 of (TOP-REAL p1) is non empty Relation-like [: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] -defined the carrier of (TOP-REAL p1) -valued Function-like V26([: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):]
[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty set
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),gqq,(- q)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(gqq + (- q)).| is V11() real ext-real non negative Element of REAL
|.(- q).| is V11() real ext-real non negative Element of REAL
|.gqq.| + |.(- q).| is V11() real ext-real non negative Element of REAL
|.(0. (TOP-REAL p1)).| is V11() real ext-real non negative Element of REAL
|.gqq.| + |.(0. (TOP-REAL p1)).| is V11() real ext-real non negative Element of REAL
|.gqq.| + {} is V11() real ext-real non negative Element of REAL
gqq - q is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),gqq,(- q)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(gqq - q).| is V11() real ext-real non negative Element of REAL
Ball (q,S2) is functional open V162( TOP-REAL p1) Element of bool the carrier of (TOP-REAL p1)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1) : not S2 <= |.(b₁ - q).| } is set
U2 is V11() real ext-real set
S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
S2 / 2 is V11() real ext-real Element of REAL
S2 * (2 ") is V11() real ext-real set
abs U2 is V11() real ext-real non negative Element of REAL
(S2 / 2) / (abs U2) is V11() real ext-real Element of REAL
(abs U2) " is V11() real ext-real non negative set
(S2 / 2) * ((abs U2) ") is V11() real ext-real set
Ball (S1,((S2 / 2) / (abs U2))) is functional open V162( TOP-REAL p1) Element of bool the carrier of (TOP-REAL p1)
bool the carrier of (TOP-REAL p1) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1) : not (S2 / 2) / (abs U2) <= |.(b₁ - S1).| } is set
(Ball (S1,((S2 / 2) / (abs U2)))) /\ ([#] c₂) is Element of bool the carrier of c₂
the topology of (TOP-REAL p1) is non empty open Element of bool (bool the carrier of (TOP-REAL p1))
bool (bool the carrier of (TOP-REAL p1)) is non empty set
X2 is Element of bool the carrier of c₂
the topology of c₂ is non empty open Element of bool (bool the carrier of c₂)
bool (bool the carrier of c₂) is non empty set
{ |.b₁.| where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : b₁ in X2 } is set
|.S1.| is V11() real ext-real non negative Element of REAL
f is set
U is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.U.| is V11() real ext-real non negative Element of REAL
f is non empty complex-membered ext-real-membered real-membered set
|.S1.| + ((S2 / 2) / (abs U2)) is V11() real ext-real Element of REAL
U is ext-real set
U3 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.U3.| is V11() real ext-real non negative Element of REAL
U3 - S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
- S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),U3,(- S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
the U7 of (TOP-REAL p1) is non empty Relation-like [: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] -defined the carrier of (TOP-REAL p1) -valued Function-like V26([: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):]
[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty set
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),U3,(- S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(U3 - S1).| is V11() real ext-real non negative Element of REAL
|.(- S1).| is V11() real ext-real non negative Element of REAL
|.U3.| - |.(- S1).| is V11() real ext-real Element of REAL
- |.(- S1).| is V11() real ext-real non positive set
|.U3.| + (- |.(- S1).|) is V11() real ext-real set
U3 + (- S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(U3 + (- S1)).| is V11() real ext-real non negative Element of REAL
|.U3.| - |.S1.| is V11() real ext-real Element of REAL
- |.S1.| is V11() real ext-real non positive set
|.U3.| + (- |.S1.|) is V11() real ext-real set
(|.U3.| - |.S1.|) + |.S1.| is V11() real ext-real Element of REAL
((S2 / 2) / (abs U2)) + |.S1.| is V11() real ext-real Element of REAL
sup f is ext-real set
U is V11() real ext-real set
the topology of R^1 is non empty open Element of bool (bool the carrier of R^1)
bool (bool the carrier of R^1) is non empty set
M2 is open complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
T2 is Element of bool the carrier of c₂
M .: T2 is complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
T2 /\ X2 is Element of bool the carrier of c₂
f2 is Element of bool the carrier of c₂
X .: f2 is functional Element of bool the carrier of (TOP-REAL T)
S is set
dom X is non empty Element of bool the carrier of c₂
Q is set
X . Q is Relation-like Function-like set
S is Element of the carrier of c₂
M . S is V11() real ext-real Element of the carrier of R^1
U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U5 is V11() real ext-real set
U5 * U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
y is Element of the carrier of (Euclid T)
q is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U2 * S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U5 - U2 is V11() real ext-real set
- U2 is V11() real ext-real set
U5 + (- U2) is V11() real ext-real set
abs (U5 - U2) is V11() real ext-real non negative Element of REAL
|.U4.| is V11() real ext-real non negative Element of REAL
(abs (U5 - U2)) * |.U4.| is V11() real ext-real non negative Element of REAL
r2 is V11() real ext-real Element of REAL
abs r2 is V11() real ext-real non negative Element of REAL
(abs r2) * |.U4.| is V11() real ext-real non negative Element of REAL
(U5 - U2) * U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.((U5 - U2) * U4).| is V11() real ext-real non negative Element of REAL
U4 - S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
- S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),U4,(- S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
the U7 of (TOP-REAL p1) is non empty Relation-like [: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] -defined the carrier of (TOP-REAL p1) -valued Function-like V26([: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):]
[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty set
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),U4,(- S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(U4 - S1).| is V11() real ext-real non negative Element of REAL
(abs U2) * |.(U4 - S1).| is V11() real ext-real non negative Element of REAL
r3 is V11() real ext-real Element of REAL
abs r3 is V11() real ext-real non negative Element of REAL
(abs r3) * |.(U4 - S1).| is V11() real ext-real non negative Element of REAL
U2 * (U4 - S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(U2 * (U4 - S1)).| is V11() real ext-real non negative Element of REAL
(abs U2) * ((S2 / 2) / (abs U2)) is V11() real ext-real Element of REAL
(abs U2) / (abs U2) is V11() real ext-real non negative Element of REAL
(abs U2) * ((abs U2) ") is V11() real ext-real non negative set
(S2 / 2) / ((abs U2) / (abs U2)) is V11() real ext-real Element of REAL
((abs U2) / (abs U2)) " is V11() real ext-real non negative set
(S2 / 2) * (((abs U2) / (abs U2)) ") is V11() real ext-real set
(S2 / 2) / 1 is V11() real ext-real Element of REAL
1 " is non empty V11() real ext-real positive non negative set
(S2 / 2) * (1 ") is V11() real ext-real set
(S2 / 2) + (S2 / 2) is V11() real ext-real Element of REAL
|.((U5 - U2) * U4).| + |.(U2 * (U4 - S1)).| is V11() real ext-real non negative Element of REAL
((U5 - U2) * U4) + (U2 * (U4 - S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),((U5 - U2) * U4),(U2 * (U4 - S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(((U5 - U2) * U4) + (U2 * (U4 - S1))).| is V11() real ext-real non negative Element of REAL
U2 * U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
(U5 * U4) - (U2 * U4) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
- (U2 * U4) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),(U5 * U4),(- (U2 * U4))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),(U5 * U4),(- (U2 * U4))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
((U5 * U4) - (U2 * U4)) + (U2 * (U4 - S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),((U5 * U4) - (U2 * U4)),(U2 * (U4 - S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
(U2 * U4) - (U2 * S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
- (U2 * S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),(U2 * U4),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),(U2 * U4),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
((U5 * U4) - (U2 * U4)) + ((U2 * U4) - (U2 * S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),((U5 * U4) - (U2 * U4)),((U2 * U4) - (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
((U5 * U4) - (U2 * U4)) + (U2 * U4) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),((U5 * U4) - (U2 * U4)),(U2 * U4)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
(((U5 * U4) - (U2 * U4)) + (U2 * U4)) - (U2 * S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),(((U5 * U4) - (U2 * U4)) + (U2 * U4)),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),(((U5 * U4) - (U2 * U4)) + (U2 * U4)),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
(U5 * U4) - (U2 * S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),(U5 * U4),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),(U5 * U4),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
gqq is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
Ball (q,S2) is functional open V162( TOP-REAL p1) Element of bool the carrier of (TOP-REAL p1)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1) : not S2 <= |.(b₁ - q).| } is set
(S2 / 2) / U is V11() real ext-real Element of REAL
U " is V11() real ext-real set
(S2 / 2) * (U ") is V11() real ext-real set
U2 - ((S2 / 2) / U) is V11() real ext-real Element of REAL
- ((S2 / 2) / U) is V11() real ext-real set
U2 + (- ((S2 / 2) / U)) is V11() real ext-real set
U2 + ((S2 / 2) / U) is V11() real ext-real Element of REAL
].(U2 - ((S2 / 2) / U)),(U2 + ((S2 / 2) / U)).[ is non left_end non right_end V317() set
((S2 / 2) / U) + U2 is V11() real ext-real Element of REAL
{} + U2 is V11() real ext-real set
- ((S2 / 2) / U) is V11() real ext-real Element of REAL
(- ((S2 / 2) / U)) + U2 is V11() real ext-real Element of REAL
M2 is open complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
T2 is Element of bool the carrier of c₂
M .: T2 is complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
T2 /\ X2 is Element of bool the carrier of c₂
f2 is Element of bool the carrier of c₂
X .: f2 is functional Element of bool the carrier of (TOP-REAL T)
S is set
dom X is non empty Element of bool the carrier of c₂
Q is set
X . Q is Relation-like Function-like set
S is Element of the carrier of c₂
M . S is V11() real ext-real Element of the carrier of R^1
U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U5 is V11() real ext-real set
U5 * U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
y is Element of the carrier of (Euclid T)
q is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U2 * S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
U5 - U2 is V11() real ext-real set
- U2 is V11() real ext-real set
U5 + (- U2) is V11() real ext-real set
abs (U5 - U2) is V11() real ext-real non negative Element of REAL
|.U4.| is V11() real ext-real non negative Element of REAL
(abs (U5 - U2)) * |.U4.| is V11() real ext-real non negative Element of REAL
r2 is V11() real ext-real Element of REAL
abs r2 is V11() real ext-real non negative Element of REAL
(abs r2) * |.U4.| is V11() real ext-real non negative Element of REAL
(U5 - U2) * U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.((U5 - U2) * U4).| is V11() real ext-real non negative Element of REAL
dom M is non empty Element of bool the carrier of c₂
((S2 / 2) / U) * U is V11() real ext-real Element of REAL
(abs (U5 - U2)) * U is V11() real ext-real Element of REAL
U / U is V11() real ext-real Element of COMPLEX
U * (U ") is V11() real ext-real set
(S2 / 2) / (U / U) is V11() real ext-real Element of REAL
(U / U) " is V11() real ext-real set
(S2 / 2) * ((U / U) ") is V11() real ext-real set
(S2 / 2) / 1 is V11() real ext-real Element of REAL
1 " is non empty V11() real ext-real positive non negative set
(S2 / 2) * (1 ") is V11() real ext-real set
|.U4.| * (abs (U5 - U2)) is V11() real ext-real non negative Element of REAL
U * (abs (U5 - U2)) is V11() real ext-real Element of REAL
U4 - S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
- S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),U4,(- S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
the U7 of (TOP-REAL p1) is non empty Relation-like [: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] -defined the carrier of (TOP-REAL p1) -valued Function-like V26([: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):]
[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL p1), the carrier of (TOP-REAL p1):], the carrier of (TOP-REAL p1):] is non empty set
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),U4,(- S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(U4 - S1).| is V11() real ext-real non negative Element of REAL
(abs U2) * |.(U4 - S1).| is V11() real ext-real non negative Element of REAL
r3 is V11() real ext-real Element of REAL
abs r3 is V11() real ext-real non negative Element of REAL
(abs r3) * |.(U4 - S1).| is V11() real ext-real non negative Element of REAL
U2 * (U4 - S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(U2 * (U4 - S1)).| is V11() real ext-real non negative Element of REAL
(abs U2) * ((S2 / 2) / (abs U2)) is V11() real ext-real Element of REAL
(abs U2) / (abs U2) is V11() real ext-real non negative Element of REAL
(abs U2) * ((abs U2) ") is V11() real ext-real non negative set
(S2 / 2) / ((abs U2) / (abs U2)) is V11() real ext-real Element of REAL
((abs U2) / (abs U2)) " is V11() real ext-real non negative set
(S2 / 2) * (((abs U2) / (abs U2)) ") is V11() real ext-real set
(S2 / 2) + (S2 / 2) is V11() real ext-real Element of REAL
|.((U5 - U2) * U4).| + |.(U2 * (U4 - S1)).| is V11() real ext-real non negative Element of REAL
((U5 - U2) * U4) + (U2 * (U4 - S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),((U5 - U2) * U4),(U2 * (U4 - S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
|.(((U5 - U2) * U4) + (U2 * (U4 - S1))).| is V11() real ext-real non negative Element of REAL
U2 * U4 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
(U5 * U4) - (U2 * U4) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
- (U2 * U4) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),(U5 * U4),(- (U2 * U4))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),(U5 * U4),(- (U2 * U4))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
((U5 * U4) - (U2 * U4)) + (U2 * (U4 - S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),((U5 * U4) - (U2 * U4)),(U2 * (U4 - S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
(U2 * U4) - (U2 * S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
- (U2 * S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),(U2 * U4),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),(U2 * U4),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
((U5 * U4) - (U2 * U4)) + ((U2 * U4) - (U2 * S1)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),((U5 * U4) - (U2 * U4)),((U2 * U4) - (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
((U5 * U4) - (U2 * U4)) + (U2 * U4) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),((U5 * U4) - (U2 * U4)),(U2 * U4)) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
(((U5 * U4) - (U2 * U4)) + (U2 * U4)) - (U2 * S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),(((U5 * U4) - (U2 * U4)) + (U2 * U4)),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),(((U5 * U4) - (U2 * U4)) + (U2 * U4)),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
(U5 * U4) - (U2 * S1) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K184((TOP-REAL p1),(U5 * U4),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
K155( the carrier of (TOP-REAL p1), the U7 of (TOP-REAL p1),(U5 * U4),(- (U2 * S1))) is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
gqq is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
Ball (q,S2) is functional open V162( TOP-REAL p1) Element of bool the carrier of (TOP-REAL p1)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1) : not S2 <= |.(b₁ - q).| } is set
U2 is V11() real ext-real set
X1 is Element of the carrier of c₂
p is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
M . X1 is V11() real ext-real Element of the carrier of R^1
p1 is V11() real ext-real set
X . p is Relation-like Function-like set
p1 * p is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
bool the carrier of (TOP-REAL T) is non empty set
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
bool the carrier of (TOP-REAL T) is non empty set
the Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
the V11() real ext-real set is V11() real ext-real set
Ball ( the Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T), the V11() real ext-real set ) is functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not the V11() real ext-real set <= |.(b₁ - the Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)).| } is set
c₂ is functional Element of bool the carrier of (TOP-REAL T)
M is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
X is V11() real ext-real set
Ball (M,X) is functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not X <= |.(b₁ - M).| } is set
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
bool the carrier of (TOP-REAL T) is non empty set
0. (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V54( TOP-REAL T) V124() V125() V126() Element of the carrier of (TOP-REAL T)
the ZeroF of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
Ball ((0. (TOP-REAL T)),1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not 1 <= |.(b₁ - (0. (TOP-REAL T))).| } is set
c₂ is functional open (T) Element of bool the carrier of (TOP-REAL T)
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
bool the carrier of (TOP-REAL T) is non empty set
c₂ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
M is functional open Element of bool the carrier of (TOP-REAL T)
the topology of (TOP-REAL T) is non empty open Element of bool (bool the carrier of (TOP-REAL T))
bool (bool the carrier of (TOP-REAL T)) is non empty set
TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #) is non empty strict TopSpace-like V204() second-countable TopStruct
Euclid T is non empty strict Reflexive discerning V83() triangle MetrStruct
TopSpaceMetr (Euclid T) is metrizable first-countable TopStruct
the carrier of (Euclid T) is non empty set
Family_open_set (Euclid T) is Element of bool (bool the carrier of (Euclid T))
bool the carrier of (Euclid T) is non empty set
bool (bool the carrier of (Euclid T)) is non empty set
TopStruct(# the carrier of (Euclid T),(Family_open_set (Euclid T)) #) is non empty strict TopStruct
X is Element of the carrier of (Euclid T)
X1 is V11() real ext-real Element of REAL
Ball (X,X1) is Element of bool the carrier of (Euclid T)
p is non empty V11() real ext-real positive non negative set
Ball (c₂,p) is non empty functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not p <= |.(b₁ - c₂).| } is set
U1 is functional open (T) Element of bool the carrier of (TOP-REAL T)
p1 is ordinal natural V11() real ext-real non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of NAT
TOP-REAL p1 is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable L15()
the carrier of (TOP-REAL p1) is non empty functional set
Ball (X,p) is Element of bool the carrier of (Euclid T)
S1 is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1)
Ball (S1,p) is non empty functional open V162( TOP-REAL p1) Element of bool the carrier of (TOP-REAL p1)
bool the carrier of (TOP-REAL p1) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p1) : not p <= |.(b₁ - S1).| } is set
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
c₂ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
M is V11() real ext-real set
Ball (c₂,M) is functional open Element of bool the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not M <= |.(b₁ - c₂).| } is set
(TOP-REAL T) | (Ball (c₂,M)) is strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
0. (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V54( TOP-REAL T) V124() V125() V126() Element of the carrier of (TOP-REAL T)
the carrier of (TOP-REAL T) is non empty functional set
the ZeroF of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(T,(0. (TOP-REAL T)),1) is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball ((0. (TOP-REAL T)),1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not 1 <= |.(b₁ - (0. (TOP-REAL T))).| } is set
(TOP-REAL T) | (Ball ((0. (TOP-REAL T)),1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
T is ordinal natural V11() real ext-real non negative V35() cardinal set
(T) is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
0. (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V54( TOP-REAL T) V124() V125() V126() Element of the carrier of (TOP-REAL T)
the carrier of (TOP-REAL T) is non empty functional set
the ZeroF of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(T,(0. (TOP-REAL T)),1) is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball ((0. (TOP-REAL T)),1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not 1 <= |.(b₁ - (0. (TOP-REAL T))).| } is set
(TOP-REAL T) | (Ball ((0. (TOP-REAL T)),1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
c₂ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
M is non empty V11() real ext-real positive non negative set
(T,c₂,M) is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball (c₂,M) is non empty functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not M <= |.(b₁ - c₂).| } is set
(TOP-REAL T) | (Ball (c₂,M)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
c₂ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
M is V11() real ext-real set
(T,c₂,M) is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball (c₂,M) is functional open Element of bool the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not M <= |.(b₁ - c₂).| } is set
(TOP-REAL T) | (Ball (c₂,M)) is strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the carrier of (T,c₂,M) is set
[#] (T,c₂,M) is non proper open closed dense Element of bool the carrier of (T,c₂,M)
bool the carrier of (T,c₂,M) is non empty set
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
(T) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
0. (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V54( TOP-REAL T) V124() V125() V126() Element of the carrier of (TOP-REAL T)
the ZeroF of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(T,(0. (TOP-REAL T)),1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball ((0. (TOP-REAL T)),1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not 1 <= |.(b₁ - (0. (TOP-REAL T))).| } is set
(TOP-REAL T) | (Ball ((0. (TOP-REAL T)),1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the carrier of (T) is non empty set
c₂ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
|.c₂.| is V11() real ext-real non negative Element of REAL
X is non empty ordinal natural V11() real ext-real positive non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of NAT
TOP-REAL X is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable L15()
0. (TOP-REAL X) is Relation-like NAT -defined Function-like V35() X -element V43() V44() V54( TOP-REAL X) V124() V125() V126() Element of the carrier of (TOP-REAL X)
the carrier of (TOP-REAL X) is non empty functional set
the ZeroF of (TOP-REAL X) is Relation-like NAT -defined Function-like V35() X -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X)
M is V11() real ext-real set
(X,(0. (TOP-REAL X)),M) is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL X
Ball ((0. (TOP-REAL X)),M) is functional open V162( TOP-REAL X) Element of bool the carrier of (TOP-REAL X)
bool the carrier of (TOP-REAL X) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() X -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X) : not M <= |.(b₁ - (0. (TOP-REAL X))).| } is set
(TOP-REAL X) | (Ball ((0. (TOP-REAL X)),M)) is strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL X
the carrier of (X,(0. (TOP-REAL X)),M) is set
Ball ((0. (TOP-REAL X)),1) is non empty functional open V162( TOP-REAL X) Element of bool the carrier of (TOP-REAL X)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() X -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X) : not 1 <= |.(b₁ - (0. (TOP-REAL X))).| } is set
T is ordinal natural V11() real ext-real non negative V35() cardinal set
(T) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
0. (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V54( TOP-REAL T) V124() V125() V126() Element of the carrier of (TOP-REAL T)
the carrier of (TOP-REAL T) is non empty functional set
the ZeroF of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(T,(0. (TOP-REAL T)),1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball ((0. (TOP-REAL T)),1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not 1 <= |.(b₁ - (0. (TOP-REAL T))).| } is set
(TOP-REAL T) | (Ball ((0. (TOP-REAL T)),1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the carrier of (T) is non empty set
[: the carrier of (T), the carrier of (TOP-REAL T):] is non empty Relation-like set
bool [: the carrier of (T), the carrier of (TOP-REAL T):] is non empty set
c₂ is non empty Relation-like the carrier of (T) -defined the carrier of (TOP-REAL T) -valued Function-like V26( the carrier of (T)) quasi_total Element of bool [: the carrier of (T), the carrier of (TOP-REAL T):]
dom c₂ is non empty Element of bool the carrier of (T)
bool the carrier of (T) is non empty set
[#] (T) is non empty non proper open closed dense non boundary Element of bool the carrier of (T)
[#] (TOP-REAL T) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL T)
M is non empty ordinal natural V11() real ext-real positive non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of NAT
TOP-REAL M is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable L15()
[#] (TOP-REAL M) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL M)
the carrier of (TOP-REAL M) is non empty functional set
bool the carrier of (TOP-REAL M) is non empty set
X is set
X1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.X1.| is V11() real ext-real non negative Element of REAL
0. (TOP-REAL M) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V54( TOP-REAL M) V124() V125() V126() Element of the carrier of (TOP-REAL M)
the ZeroF of (TOP-REAL M) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
X1 - (0. (TOP-REAL M)) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
- (0. (TOP-REAL M)) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
K184((TOP-REAL M),X1,(- (0. (TOP-REAL M)))) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
the U7 of (TOP-REAL M) is non empty Relation-like [: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):] -defined the carrier of (TOP-REAL M) -valued Function-like V26([: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):]
[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):] is non empty set
K155( the carrier of (TOP-REAL M), the U7 of (TOP-REAL M),X1,(- (0. (TOP-REAL M)))) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.(X1 - (0. (TOP-REAL M))).| is V11() real ext-real non negative Element of REAL
[X1,X] is set
{X1,X} is non empty set
{X1} is non empty trivial functional 1 -element set
{{X1,X},{X1}} is non empty set
c₂ . X1 is Relation-like Function-like set
|.X1.| * |.X1.| is V11() real ext-real non negative Element of REAL
1 - (|.X1.| * |.X1.|) is V11() real ext-real Element of REAL
- (|.X1.| * |.X1.|) is V11() real ext-real non positive set
1 + (- (|.X1.| * |.X1.|)) is V11() real ext-real set
1 / (1 - (|.X1.| * |.X1.|)) is V11() real ext-real Element of REAL
(1 - (|.X1.| * |.X1.|)) " is V11() real ext-real set
1 * ((1 - (|.X1.| * |.X1.|)) ") is V11() real ext-real set
(1 / (1 - (|.X1.| * |.X1.|))) * X1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
X1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.X1.| is V11() real ext-real non negative Element of REAL
|.X1.| * |.X1.| is V11() real ext-real non negative Element of REAL
4 is non empty ordinal natural V11() real ext-real positive non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of NAT
4 * (|.X1.| * |.X1.|) is V11() real ext-real non negative Element of REAL
1 + (4 * (|.X1.| * |.X1.|)) is non empty V11() real ext-real positive non negative Element of REAL
sqrt (1 + (4 * (|.X1.| * |.X1.|))) is V11() real ext-real Element of REAL
1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) " is V11() real ext-real set
2 * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
(2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).| is V11() real ext-real non negative Element of REAL
U1 is V11() real ext-real Element of REAL
abs U1 is V11() real ext-real non negative Element of REAL
(abs U1) * |.X1.| is V11() real ext-real non negative Element of REAL
U1 * |.X1.| is V11() real ext-real Element of REAL
|.X1.| * 4 is V11() real ext-real non negative Element of REAL
(|.X1.| * 4) + (1 + (4 * (|.X1.| * |.X1.|))) is non empty V11() real ext-real positive non negative Element of REAL
{} + (1 + (4 * (|.X1.| * |.X1.|))) is non empty V11() real ext-real positive non negative Element of REAL
2 * |.X1.| is V11() real ext-real non negative Element of REAL
1 + (2 * |.X1.|) is non empty V11() real ext-real positive non negative Element of REAL
(1 + (2 * |.X1.|)) ^2 is V11() real ext-real Element of REAL
(1 + (2 * |.X1.|)) * (1 + (2 * |.X1.|)) is non empty V11() real ext-real positive non negative set
sqrt ((1 + (2 * |.X1.|)) ^2) is V11() real ext-real Element of REAL
(1 + (2 * |.X1.|)) - (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
- (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real set
(1 + (2 * |.X1.|)) + (- (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real set
(sqrt (1 + (4 * (|.X1.| * |.X1.|)))) - (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
(sqrt (1 + (4 * (|.X1.| * |.X1.|)))) + (- (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real set
1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
1 + (- (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real set
(1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * |.X1.|) is V11() real ext-real Element of REAL
((1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * |.X1.|)) - (2 * |.X1.|) is V11() real ext-real Element of REAL
- (2 * |.X1.|) is V11() real ext-real non positive set
((1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * |.X1.|)) + (- (2 * |.X1.|)) is V11() real ext-real set
{} - (2 * |.X1.|) is V11() real ext-real non positive Element of REAL
{} + (- (2 * |.X1.|)) is V11() real ext-real non positive set
(1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * 1 is V11() real ext-real Element of REAL
- (2 * |.X1.|) is V11() real ext-real non positive Element of REAL
(1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
(1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real Element of REAL
(sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2 is V11() real ext-real Element of REAL
(sqrt (1 + (4 * (|.X1.| * |.X1.|)))) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real set
1 - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2) is V11() real ext-real Element of REAL
- ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2) is V11() real ext-real set
1 + (- ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) is V11() real ext-real set
(1 - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(1 - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
1 - (1 + (4 * (|.X1.| * |.X1.|))) is V11() real ext-real Element of REAL
- (1 + (4 * (|.X1.| * |.X1.|))) is non empty V11() real ext-real non positive negative set
1 + (- (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real set
(1 - (1 + (4 * (|.X1.| * |.X1.|)))) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(1 - (1 + (4 * (|.X1.| * |.X1.|)))) * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
(4 * (|.X1.| * |.X1.|)) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(4 * (|.X1.| * |.X1.|)) * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
- ((4 * (|.X1.| * |.X1.|)) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real Element of REAL
(2 * |.X1.|) * (2 * |.X1.|) is V11() real ext-real non negative Element of REAL
((2 * |.X1.|) * (2 * |.X1.|)) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
((2 * |.X1.|) * (2 * |.X1.|)) * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
(2 * |.X1.|) / (2 * |.X1.|) is V11() real ext-real non negative Element of REAL
(2 * |.X1.|) " is V11() real ext-real non negative set
(2 * |.X1.|) * ((2 * |.X1.|) ") is V11() real ext-real non negative set
(2 * |.X1.|) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(2 * |.X1.|) * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
(2 * |.X1.|) * ((2 * |.X1.|) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real Element of REAL
((2 * |.X1.|) * ((2 * |.X1.|) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))))) / (2 * |.X1.|) is V11() real ext-real Element of REAL
((2 * |.X1.|) * ((2 * |.X1.|) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))))) * ((2 * |.X1.|) ") is V11() real ext-real set
((2 * |.X1.|) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) / ((2 * |.X1.|) / (2 * |.X1.|)) is V11() real ext-real Element of REAL
((2 * |.X1.|) / (2 * |.X1.|)) " is V11() real ext-real non negative set
((2 * |.X1.|) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * (((2 * |.X1.|) / (2 * |.X1.|)) ") is V11() real ext-real set
((2 * |.X1.|) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) / 1 is V11() real ext-real Element of REAL
1 " is non empty V11() real ext-real positive non negative set
((2 * |.X1.|) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * (1 ") is V11() real ext-real set
0. (TOP-REAL M) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V54( TOP-REAL M) V124() V125() V126() Element of the carrier of (TOP-REAL M)
the ZeroF of (TOP-REAL M) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1) - (0. (TOP-REAL M)) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
- (0. (TOP-REAL M)) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
K184((TOP-REAL M),((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1),(- (0. (TOP-REAL M)))) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
the U7 of (TOP-REAL M) is non empty Relation-like [: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):] -defined the carrier of (TOP-REAL M) -valued Function-like V26([: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):]
[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):] is non empty set
K155( the carrier of (TOP-REAL M), the U7 of (TOP-REAL M),((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1),(- (0. (TOP-REAL M)))) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.(((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1) - (0. (TOP-REAL M))).| is V11() real ext-real non negative Element of REAL
Ball ((0. (TOP-REAL M)),1) is non empty functional open V162( TOP-REAL M) Element of bool the carrier of (TOP-REAL M)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M) : not 1 <= |.(b₁ - (0. (TOP-REAL M))).| } is set
[((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1),X] is set
{((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1),X} is non empty set
{((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1)} is non empty trivial functional 1 -element set
{{((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1),X},{((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1)}} is non empty set
1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
- (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real set
1 + (- (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real set
(sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2 is V11() real ext-real Element of REAL
(sqrt (1 + (4 * (|.X1.| * |.X1.|)))) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real set
|.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).| * |.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).| is V11() real ext-real non negative Element of REAL
(2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * (2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real Element of REAL
((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * (2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))))) * (|.X1.| * |.X1.|) is V11() real ext-real Element of REAL
2 * 2 is non empty ordinal natural V11() real ext-real positive non negative V35() cardinal Element of REAL
(1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(2 * 2) / ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real Element of REAL
((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) " is V11() real ext-real set
(2 * 2) * (((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) ") is V11() real ext-real set
((2 * 2) / ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))))) * (|.X1.| * |.X1.|) is V11() real ext-real Element of REAL
2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2 is V11() real ext-real Element of REAL
(sqrt (1 + (4 * (|.X1.| * |.X1.|)))) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real set
(1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2) is V11() real ext-real Element of REAL
4 / ((1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) is V11() real ext-real Element of REAL
((1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) " is V11() real ext-real set
4 * (((1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) ") is V11() real ext-real set
(4 / ((1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2))) * (|.X1.| * |.X1.|) is V11() real ext-real Element of REAL
(1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + (1 + (4 * (|.X1.| * |.X1.|))) is V11() real ext-real Element of REAL
4 / ((1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
((1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + (1 + (4 * (|.X1.| * |.X1.|)))) " is V11() real ext-real set
4 * (((1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + (1 + (4 * (|.X1.| * |.X1.|)))) ") is V11() real ext-real set
(4 / ((1 + (2 * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + (1 + (4 * (|.X1.| * |.X1.|))))) * (|.X1.| * |.X1.|) is V11() real ext-real Element of REAL
2 * (|.X1.| * |.X1.|) is V11() real ext-real non negative Element of REAL
(1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * (|.X1.| * |.X1.|)) is V11() real ext-real Element of REAL
2 * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * (|.X1.| * |.X1.|))) is V11() real ext-real Element of REAL
4 / (2 * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
(2 * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * (|.X1.| * |.X1.|)))) " is V11() real ext-real set
4 * ((2 * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * (|.X1.| * |.X1.|)))) ") is V11() real ext-real set
(4 / (2 * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * (|.X1.| * |.X1.|))))) * (|.X1.| * |.X1.|) is V11() real ext-real Element of REAL
4 / 2 is non empty V11() real ext-real positive non negative Element of REAL
4 * (2 ") is non empty V11() real ext-real positive non negative set
(4 / 2) / ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * (|.X1.| * |.X1.|))) is V11() real ext-real Element of REAL
((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * (|.X1.| * |.X1.|))) " is V11() real ext-real set
(4 / 2) * (((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * (|.X1.| * |.X1.|))) ") is V11() real ext-real set
((4 / 2) / ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) + (2 * (|.X1.| * |.X1.|)))) * (|.X1.| * |.X1.|) is V11() real ext-real Element of REAL
1 + (2 * (|.X1.| * |.X1.|)) is non empty V11() real ext-real positive non negative Element of REAL
(1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
(2 * (|.X1.| * |.X1.|)) / ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) " is V11() real ext-real set
(2 * (|.X1.| * |.X1.|)) * (((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
1 - (|.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).| * |.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).|) is V11() real ext-real Element of REAL
- (|.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).| * |.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).|) is V11() real ext-real non positive set
1 + (- (|.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).| * |.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).|)) is V11() real ext-real set
((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) / ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * (((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
- ((2 * (|.X1.| * |.X1.|)) / ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real Element of REAL
(((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) / ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + (- ((2 * (|.X1.| * |.X1.|)) / ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))))) is V11() real ext-real Element of REAL
(1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) / ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * (((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) / ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * 1 is V11() real ext-real Element of REAL
(1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) / (1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) " is V11() real ext-real set
(1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * ((1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) / ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * ((1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) / (1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real Element of REAL
(1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * (1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * (1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) / ((1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real Element of REAL
((1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) " is V11() real ext-real set
((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * (1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * (((1 - (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) * ((1 + (2 * (|.X1.| * |.X1.|))) + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) ") is V11() real ext-real set
1 - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2) is V11() real ext-real Element of REAL
- ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2) is V11() real ext-real set
1 + (- ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) is V11() real ext-real set
(2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
(1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
- ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real set
(1 + (2 * (|.X1.| * |.X1.|))) + (- ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real set
(sqrt (1 + (4 * (|.X1.| * |.X1.|)))) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) * (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
- ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) * (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real set
((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + (- ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real set
(1 - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) / (((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real Element of REAL
(((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) " is V11() real ext-real set
(1 - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) * ((((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) ") is V11() real ext-real set
1 - (1 + (4 * (|.X1.| * |.X1.|))) is V11() real ext-real Element of REAL
- (1 + (4 * (|.X1.| * |.X1.|))) is non empty V11() real ext-real non positive negative set
1 + (- (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real set
((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2) is V11() real ext-real Element of REAL
((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + (- ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) is V11() real ext-real set
(1 - (1 + (4 * (|.X1.| * |.X1.|)))) / (((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) is V11() real ext-real Element of REAL
(((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) " is V11() real ext-real set
(1 - (1 + (4 * (|.X1.| * |.X1.|)))) * ((((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - ((sqrt (1 + (4 * (|.X1.| * |.X1.|)))) ^2)) ") is V11() real ext-real set
- (4 * (|.X1.| * |.X1.|)) is V11() real ext-real non positive Element of REAL
((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - (1 + (4 * (|.X1.| * |.X1.|))) is V11() real ext-real Element of REAL
((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) + (- (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real set
(- (4 * (|.X1.| * |.X1.|))) / (((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - (1 + (4 * (|.X1.| * |.X1.|)))) is V11() real ext-real Element of REAL
(((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - (1 + (4 * (|.X1.| * |.X1.|)))) " is V11() real ext-real set
(- (4 * (|.X1.| * |.X1.|))) * ((((1 + (2 * (|.X1.| * |.X1.|))) - ((2 * (|.X1.| * |.X1.|)) * (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) - (1 + (4 * (|.X1.| * |.X1.|)))) ") is V11() real ext-real set
- (2 * (|.X1.| * |.X1.|)) is V11() real ext-real non positive Element of REAL
(- (2 * (|.X1.| * |.X1.|))) * (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(- (4 * (|.X1.| * |.X1.|))) / ((- (2 * (|.X1.| * |.X1.|))) * (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) is V11() real ext-real Element of REAL
((- (2 * (|.X1.| * |.X1.|))) * (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) " is V11() real ext-real set
(- (4 * (|.X1.| * |.X1.|))) * (((- (2 * (|.X1.| * |.X1.|))) * (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) ") is V11() real ext-real set
2 * (- (2 * (|.X1.| * |.X1.|))) is V11() real ext-real non positive Element of REAL
(2 * (- (2 * (|.X1.| * |.X1.|)))) / (- (2 * (|.X1.| * |.X1.|))) is V11() real ext-real non negative Element of REAL
(- (2 * (|.X1.| * |.X1.|))) " is V11() real ext-real non positive set
(2 * (- (2 * (|.X1.| * |.X1.|)))) * ((- (2 * (|.X1.| * |.X1.|))) ") is V11() real ext-real non negative set
((2 * (- (2 * (|.X1.| * |.X1.|)))) / (- (2 * (|.X1.| * |.X1.|)))) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
((2 * (- (2 * (|.X1.| * |.X1.|)))) / (- (2 * (|.X1.| * |.X1.|)))) * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
(- (2 * (|.X1.| * |.X1.|))) / (- (2 * (|.X1.| * |.X1.|))) is V11() real ext-real non negative Element of REAL
(- (2 * (|.X1.| * |.X1.|))) * ((- (2 * (|.X1.| * |.X1.|))) ") is V11() real ext-real non negative set
2 * ((- (2 * (|.X1.| * |.X1.|))) / (- (2 * (|.X1.| * |.X1.|)))) is V11() real ext-real non negative Element of REAL
(2 * ((- (2 * (|.X1.| * |.X1.|))) / (- (2 * (|.X1.| * |.X1.|))))) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(2 * ((- (2 * (|.X1.| * |.X1.|))) / (- (2 * (|.X1.| * |.X1.|))))) * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
2 * 1 is non empty ordinal natural V11() real ext-real positive non negative V35() cardinal Element of REAL
(2 * 1) / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) is V11() real ext-real Element of REAL
(2 * 1) * ((1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|))))) ") is V11() real ext-real set
c₂ . ((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1) is Relation-like Function-like set
1 / (1 - (|.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).| * |.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).|)) is V11() real ext-real Element of REAL
(1 - (|.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).| * |.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).|)) " is V11() real ext-real set
1 * ((1 - (|.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).| * |.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).|)) ") is V11() real ext-real set
(1 / (1 - (|.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).| * |.((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1).|))) * ((2 / (1 + (sqrt (1 + (4 * (|.X1.| * |.X1.|)))))) * X1) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
U1 / U1 is V11() real ext-real Element of REAL
U1 " is V11() real ext-real set
U1 * (U1 ") is V11() real ext-real set
(U1 / U1) * X1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
1 * X1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
X1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.X1.| is V11() real ext-real non negative Element of REAL
rng c₂ is non empty functional Element of bool the carrier of (TOP-REAL T)
X is set
X1 is set
c₂ . X is Relation-like Function-like set
c₂ . X1 is Relation-like Function-like set
p is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.p.| is V11() real ext-real non negative Element of REAL
|.p.| * |.p.| is V11() real ext-real non negative Element of REAL
1 - (|.p.| * |.p.|) is V11() real ext-real Element of REAL
- (|.p.| * |.p.|) is V11() real ext-real non positive set
1 + (- (|.p.| * |.p.|)) is V11() real ext-real set
1 / (1 - (|.p.| * |.p.|)) is V11() real ext-real Element of REAL
(1 - (|.p.| * |.p.|)) " is V11() real ext-real set
1 * ((1 - (|.p.| * |.p.|)) ") is V11() real ext-real set
(1 / (1 - (|.p.| * |.p.|))) * p is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
p1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.p1.| is V11() real ext-real non negative Element of REAL
|.p1.| * |.p1.| is V11() real ext-real non negative Element of REAL
1 - (|.p1.| * |.p1.|) is V11() real ext-real Element of REAL
- (|.p1.| * |.p1.|) is V11() real ext-real non positive set
1 + (- (|.p1.| * |.p1.|)) is V11() real ext-real set
1 / (1 - (|.p1.| * |.p1.|)) is V11() real ext-real Element of REAL
(1 - (|.p1.| * |.p1.|)) " is V11() real ext-real set
1 * ((1 - (|.p1.| * |.p1.|)) ") is V11() real ext-real set
(1 / (1 - (|.p1.| * |.p1.|))) * p1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.p1.| ^2 is V11() real ext-real Element of REAL
|.p1.| * |.p1.| is V11() real ext-real non negative set
sqrt (|.p1.| ^2) is V11() real ext-real Element of REAL
U1 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
U1 - (0. (TOP-REAL T)) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
- (0. (TOP-REAL T)) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
K184((TOP-REAL T),U1,(- (0. (TOP-REAL T)))) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
the U7 of (TOP-REAL T) is non empty Relation-like [: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):] -defined the carrier of (TOP-REAL T) -valued Function-like V26([: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):], the carrier of (TOP-REAL T):]
[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):], the carrier of (TOP-REAL T):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):], the carrier of (TOP-REAL T):] is non empty set
K155( the carrier of (TOP-REAL T), the U7 of (TOP-REAL T),U1,(- (0. (TOP-REAL T)))) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
|.(U1 - (0. (TOP-REAL T))).| is V11() real ext-real non negative Element of REAL
1 * |.p.| is V11() real ext-real non negative Element of REAL
- (|.p.| * |.p.|) is V11() real ext-real non positive Element of REAL
- 1 is non empty V11() real ext-real non positive negative Element of REAL
(- (|.p.| * |.p.|)) + 1 is V11() real ext-real Element of REAL
(- 1) + 1 is V11() real ext-real Element of REAL
1 * p is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
(1 - (|.p.| * |.p.|)) / (1 - (|.p.| * |.p.|)) is V11() real ext-real Element of REAL
(1 - (|.p.| * |.p.|)) * ((1 - (|.p.| * |.p.|)) ") is V11() real ext-real set
((1 - (|.p.| * |.p.|)) / (1 - (|.p.| * |.p.|))) * p is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
(1 - (|.p.| * |.p.|)) * ((1 / (1 - (|.p.| * |.p.|))) * p) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
(1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)) is V11() real ext-real Element of REAL
(1 - (|.p.| * |.p.|)) * ((1 - (|.p1.| * |.p1.|)) ") is V11() real ext-real set
((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * p1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
0. (TOP-REAL M) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V54( TOP-REAL M) V124() V125() V126() Element of the carrier of (TOP-REAL M)
the ZeroF of (TOP-REAL M) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
U1 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
U1 - (0. (TOP-REAL T)) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
- (0. (TOP-REAL T)) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
K184((TOP-REAL T),U1,(- (0. (TOP-REAL T)))) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
the U7 of (TOP-REAL T) is non empty Relation-like [: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):] -defined the carrier of (TOP-REAL T) -valued Function-like V26([: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):], the carrier of (TOP-REAL T):]
[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):], the carrier of (TOP-REAL T):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):], the carrier of (TOP-REAL T):] is non empty set
K155( the carrier of (TOP-REAL T), the U7 of (TOP-REAL T),U1,(- (0. (TOP-REAL T)))) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
|.(U1 - (0. (TOP-REAL T))).| is V11() real ext-real non negative Element of REAL
1 * |.p1.| is V11() real ext-real non negative Element of REAL
- (|.p1.| * |.p1.|) is V11() real ext-real non positive Element of REAL
(- (|.p1.| * |.p1.|)) + 1 is V11() real ext-real Element of REAL
(1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) is V11() real ext-real Element of REAL
((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * p1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
((1 / (1 - (|.p1.| * |.p1.|))) * p1) - (((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * p1) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
- (((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * p1) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
K184((TOP-REAL M),((1 / (1 - (|.p1.| * |.p1.|))) * p1),(- (((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * p1))) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
the U7 of (TOP-REAL M) is non empty Relation-like [: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):] -defined the carrier of (TOP-REAL M) -valued Function-like V26([: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):]
[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):] is non empty set
K155( the carrier of (TOP-REAL M), the U7 of (TOP-REAL M),((1 / (1 - (|.p1.| * |.p1.|))) * p1),(- (((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * p1))) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
- ((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) is V11() real ext-real Element of REAL
(- ((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))))) * p1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
((1 / (1 - (|.p1.| * |.p1.|))) * p1) + ((- ((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))))) * p1) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
K155( the carrier of (TOP-REAL M), the U7 of (TOP-REAL M),((1 / (1 - (|.p1.| * |.p1.|))) * p1),((- ((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))))) * p1)) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
(1 / (1 - (|.p1.| * |.p1.|))) + (- ((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))))) is V11() real ext-real Element of REAL
((1 / (1 - (|.p1.| * |.p1.|))) + (- ((1 / (1 - (|.p.| * |.p.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))))) * p1 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) is V11() real ext-real Element of REAL
abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) is V11() real ext-real non negative Element of REAL
(abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * (abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) is V11() real ext-real non negative Element of REAL
- ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) is V11() real ext-real Element of REAL
- ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) is V11() real ext-real Element of REAL
1 * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) is V11() real ext-real Element of REAL
(1 * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) / (1 - (|.p.| * |.p.|)) is V11() real ext-real Element of REAL
(1 * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * ((1 - (|.p.| * |.p.|)) ") is V11() real ext-real set
(1 - (|.p.| * |.p.|)) / ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) is V11() real ext-real Element of REAL
((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) " is V11() real ext-real set
(1 - (|.p.| * |.p.|)) * (((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) ") is V11() real ext-real set
1 / ((1 - (|.p.| * |.p.|)) / ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) is V11() real ext-real Element of REAL
((1 - (|.p.| * |.p.|)) / ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) " is V11() real ext-real set
1 * (((1 - (|.p.| * |.p.|)) / ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) ") is V11() real ext-real set
((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * (1 - (|.p1.| * |.p1.|)) is V11() real ext-real Element of REAL
((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) / ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) is V11() real ext-real Element of REAL
((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * (((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) ") is V11() real ext-real set
(1 - (|.p.| * |.p.|)) / (((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) / ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) is V11() real ext-real Element of REAL
(((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) / ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) " is V11() real ext-real set
(1 - (|.p.| * |.p.|)) * ((((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) / ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) ") is V11() real ext-real set
(1 - (|.p.| * |.p.|)) / 1 is V11() real ext-real Element of REAL
1 " is non empty V11() real ext-real positive non negative set
(1 - (|.p.| * |.p.|)) * (1 ") is V11() real ext-real set
(abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.| is V11() real ext-real non negative Element of REAL
|.(((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * p1).| is V11() real ext-real non negative Element of REAL
((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * |.(((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * p1).| is V11() real ext-real non negative Element of REAL
1 - (((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * |.(((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * p1).|) is V11() real ext-real Element of REAL
- (((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * |.(((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * p1).|) is V11() real ext-real non positive set
1 + (- (((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * |.(((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * p1).|)) is V11() real ext-real set
((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * ((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) is V11() real ext-real non negative Element of REAL
1 - (((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * ((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|)) is V11() real ext-real Element of REAL
- (((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * ((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|)) is V11() real ext-real non positive set
1 + (- (((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * ((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|))) is V11() real ext-real set
((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * (abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))))) * |.p1.| is V11() real ext-real non negative Element of REAL
(((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * (abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))))) * |.p1.|) * |.p1.| is V11() real ext-real non negative Element of REAL
1 - ((((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * (abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))))) * |.p1.|) * |.p1.|) is V11() real ext-real Element of REAL
- ((((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * (abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))))) * |.p1.|) * |.p1.|) is V11() real ext-real non positive set
1 + (- ((((abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * (abs ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))))) * |.p1.|) * |.p1.|)) is V11() real ext-real set
(((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.| is V11() real ext-real Element of REAL
((((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * |.p1.| is V11() real ext-real Element of REAL
1 - (((((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * |.p1.|) is V11() real ext-real Element of REAL
- (((((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * |.p1.|) is V11() real ext-real set
1 + (- (((((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) * |.p1.|) * |.p1.|)) is V11() real ext-real set
((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * |.p1.| is V11() real ext-real Element of REAL
(((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * |.p1.|) * |.p1.| is V11() real ext-real Element of REAL
1 + ((((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * |.p1.|) * |.p1.|) is V11() real ext-real Element of REAL
1 - ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) is V11() real ext-real Element of REAL
- ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) is V11() real ext-real set
1 + (- ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) is V11() real ext-real set
(1 + ((((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|))) * |.p1.|) * |.p1.|)) * (1 - ((1 - (|.p.| * |.p.|)) / (1 - (|.p1.| * |.p1.|)))) is V11() real ext-real Element of REAL
[: the carrier of (TOP-REAL M), the carrier of R^1:] is non empty Relation-like V124() V125() V126() set
bool [: the carrier of (TOP-REAL M), the carrier of R^1:] is non empty set
X is non empty Relation-like the carrier of (TOP-REAL M) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL M)) quasi_total V124() V125() V126() Element of bool [: the carrier of (TOP-REAL M), the carrier of R^1:]
X1 is non empty Relation-like the carrier of (TOP-REAL M) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL M)) quasi_total V124() V125() V126() Element of bool [: the carrier of (TOP-REAL M), the carrier of R^1:]
[: the carrier of (TOP-REAL T), the carrier of R^1:] is non empty Relation-like V124() V125() V126() set
bool [: the carrier of (TOP-REAL T), the carrier of R^1:] is non empty set
p is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total V124() V125() V126() Element of bool [: the carrier of (TOP-REAL T), the carrier of R^1:]
p1 is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total V124() V125() V126() Element of bool [: the carrier of (TOP-REAL T), the carrier of R^1:]
U1 is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total continuous V124() V125() V126() Element of bool [: the carrier of (TOP-REAL T), the carrier of R^1:]
S1 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
U1 . S1 is V11() real ext-real Element of the carrier of R^1
|.S1.| is V11() real ext-real non negative Element of REAL
|.S1.| * |.S1.| is V11() real ext-real non negative Element of REAL
1 - (|.S1.| * |.S1.|) is V11() real ext-real Element of REAL
- (|.S1.| * |.S1.|) is V11() real ext-real non positive set
1 + (- (|.S1.| * |.S1.|)) is V11() real ext-real set
X . S1 is V11() real ext-real set
p . S1 is V11() real ext-real Element of the carrier of R^1
(X . S1) - (p . S1) is V11() real ext-real set
- (p . S1) is V11() real ext-real set
(X . S1) + (- (p . S1)) is V11() real ext-real set
1 - (p . S1) is V11() real ext-real Element of REAL
1 + (- (p . S1)) is V11() real ext-real set
X1 . S1 is V11() real ext-real set
(X1 . S1) * (X1 . S1) is V11() real ext-real set
1 - ((X1 . S1) * (X1 . S1)) is V11() real ext-real Element of REAL
- ((X1 . S1) * (X1 . S1)) is V11() real ext-real set
1 + (- ((X1 . S1) * (X1 . S1))) is V11() real ext-real set
|.S1.| * (X1 . S1) is V11() real ext-real Element of REAL
1 - (|.S1.| * (X1 . S1)) is V11() real ext-real Element of REAL
- (|.S1.| * (X1 . S1)) is V11() real ext-real set
1 + (- (|.S1.| * (X1 . S1))) is V11() real ext-real set
U1 | (T) is non empty Relation-like the carrier of (T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (T)) quasi_total continuous V124() V125() V126() Element of bool [: the carrier of (T), the carrier of R^1:]
[: the carrier of (T), the carrier of R^1:] is non empty Relation-like V124() V125() V126() set
bool [: the carrier of (T), the carrier of R^1:] is non empty set
K62( the carrier of (TOP-REAL T), the carrier of R^1,U1, the carrier of (T)) is Relation-like the carrier of (TOP-REAL T) -defined the carrier of (T) -defined the carrier of (TOP-REAL T) -defined the carrier of R^1 -valued Function-like V124() V125() V126() Element of bool [: the carrier of (TOP-REAL T), the carrier of R^1:]
U2 is Element of the carrier of (T)
(U1 | (T)) . U2 is V11() real ext-real Element of the carrier of R^1
U1 . U2 is V11() real ext-real set
S2 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.S2.| is V11() real ext-real non negative Element of REAL
|.S2.| * |.S2.| is V11() real ext-real non negative Element of REAL
1 - (|.S2.| * |.S2.|) is V11() real ext-real Element of REAL
- (|.S2.| * |.S2.|) is V11() real ext-real non positive set
1 + (- (|.S2.| * |.S2.|)) is V11() real ext-real set
1 * |.S2.| is V11() real ext-real non negative Element of REAL
U2 is non empty Relation-like the carrier of (T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (T)) quasi_total V124() V125() V126() Element of bool [: the carrier of (T), the carrier of R^1:]
S2 is non empty Relation-like the carrier of (T) -defined the carrier of (TOP-REAL T) -valued Function-like V26( the carrier of (T)) quasi_total Element of bool [: the carrier of (T), the carrier of (TOP-REAL T):]
dom S2 is non empty Element of bool the carrier of (T)
X2 is set
c₂ . X2 is Relation-like Function-like set
S2 . X2 is Relation-like Function-like set
f is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(U1 | (T)) . f is V11() real ext-real set
U1 . f is V11() real ext-real Element of the carrier of R^1
|.f.| is V11() real ext-real non negative Element of REAL
|.f.| * |.f.| is V11() real ext-real non negative Element of REAL
1 - (|.f.| * |.f.|) is V11() real ext-real Element of REAL
- (|.f.| * |.f.|) is V11() real ext-real non positive set
1 + (- (|.f.| * |.f.|)) is V11() real ext-real set
U is Element of the carrier of (T)
U2 . U is V11() real ext-real Element of the carrier of R^1
1 / (1 - (|.f.| * |.f.|)) is V11() real ext-real Element of REAL
(1 - (|.f.| * |.f.|)) " is V11() real ext-real set
1 * ((1 - (|.f.| * |.f.|)) ") is V11() real ext-real set
S2 . U is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(1 / (1 - (|.f.| * |.f.|))) * f is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
4 is non empty ordinal natural V11() real ext-real positive non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of NAT
X2 is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total V124() V125() V126() Element of bool [: the carrier of (TOP-REAL T), the carrier of R^1:]
U is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total V124() V125() V126() Element of bool [: the carrier of (TOP-REAL T), the carrier of R^1:]
f is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
U . f is V11() real ext-real Element of the carrier of R^1
|.f.| is V11() real ext-real non negative Element of REAL
4 * |.f.| is V11() real ext-real non negative Element of REAL
(4 * |.f.|) * |.f.| is V11() real ext-real non negative Element of REAL
1 + ((4 * |.f.|) * |.f.|) is non empty V11() real ext-real positive non negative Element of REAL
X . f is V11() real ext-real set
X2 . f is V11() real ext-real Element of the carrier of R^1
(X . f) + (X2 . f) is V11() real ext-real set
1 + (X2 . f) is V11() real ext-real Element of REAL
p . f is V11() real ext-real Element of the carrier of R^1
4 * (p . f) is V11() real ext-real Element of REAL
1 + (4 * (p . f)) is V11() real ext-real Element of REAL
X1 . f is V11() real ext-real set
(X1 . f) * (X1 . f) is V11() real ext-real set
4 * ((X1 . f) * (X1 . f)) is V11() real ext-real Element of REAL
1 + (4 * ((X1 . f) * (X1 . f))) is V11() real ext-real Element of REAL
(X1 . f) * |.f.| is V11() real ext-real Element of REAL
4 * ((X1 . f) * |.f.|) is V11() real ext-real Element of REAL
1 + (4 * ((X1 . f) * |.f.|)) is V11() real ext-real Element of REAL
|.f.| * |.f.| is V11() real ext-real non negative Element of REAL
4 * (|.f.| * |.f.|) is V11() real ext-real non negative Element of REAL
1 + (4 * (|.f.| * |.f.|)) is non empty V11() real ext-real positive non negative Element of REAL
f is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
U . f is V11() real ext-real Element of the carrier of R^1
|.f.| is V11() real ext-real non negative Element of REAL
4 * |.f.| is V11() real ext-real non negative Element of REAL
(4 * |.f.|) * |.f.| is V11() real ext-real non negative Element of REAL
1 + ((4 * |.f.|) * |.f.|) is non empty V11() real ext-real positive non negative Element of REAL
f is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total V124() V125() V126() Element of bool [: the carrier of (TOP-REAL T), the carrier of R^1:]
U is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total V124() V125() V126() Element of bool [: the carrier of (TOP-REAL T), the carrier of R^1:]
U3 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
U . U3 is V11() real ext-real Element of the carrier of R^1
X . U3 is V11() real ext-real set
f . U3 is V11() real ext-real Element of the carrier of R^1
(X . U3) + (f . U3) is V11() real ext-real set
1 + (f . U3) is V11() real ext-real Element of REAL
U . U3 is V11() real ext-real Element of the carrier of R^1
sqrt (U . U3) is V11() real ext-real set
1 + (sqrt (U . U3)) is V11() real ext-real Element of REAL
M2 is V11() real ext-real set
U3 is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total V124() V125() V126() Element of bool [: the carrier of (TOP-REAL T), the carrier of R^1:]
M2 is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of R^1 -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total V124() V125() V126() Element of bool [: the carrier of (TOP-REAL T), the carrier of R^1:]
T2 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
M2 . T2 is V11() real ext-real Element of the carrier of R^1
|.T2.| is V11() real ext-real non negative Element of REAL
4 * |.T2.| is V11() real ext-real non negative Element of REAL
(4 * |.T2.|) * |.T2.| is V11() real ext-real non negative Element of REAL
1 + ((4 * |.T2.|) * |.T2.|) is non empty V11() real ext-real positive non negative Element of REAL
sqrt (1 + ((4 * |.T2.|) * |.T2.|)) is V11() real ext-real Element of REAL
1 + (sqrt (1 + ((4 * |.T2.|) * |.T2.|))) is V11() real ext-real Element of REAL
2 / (1 + (sqrt (1 + ((4 * |.T2.|) * |.T2.|)))) is V11() real ext-real Element of REAL
(1 + (sqrt (1 + ((4 * |.T2.|) * |.T2.|)))) " is V11() real ext-real set
2 * ((1 + (sqrt (1 + ((4 * |.T2.|) * |.T2.|)))) ") is V11() real ext-real set
U3 . T2 is V11() real ext-real Element of the carrier of R^1
2 * (U3 . T2) is V11() real ext-real Element of REAL
U . T2 is V11() real ext-real Element of the carrier of R^1
1 / (U . T2) is V11() real ext-real Element of REAL
(U . T2) " is V11() real ext-real set
1 * ((U . T2) ") is V11() real ext-real set
2 * (1 / (U . T2)) is V11() real ext-real Element of REAL
X . T2 is V11() real ext-real set
f . T2 is V11() real ext-real Element of the carrier of R^1
(X . T2) + (f . T2) is V11() real ext-real set
2 / ((X . T2) + (f . T2)) is V11() real ext-real Element of REAL
((X . T2) + (f . T2)) " is V11() real ext-real set
2 * (((X . T2) + (f . T2)) ") is V11() real ext-real set
U . T2 is V11() real ext-real Element of the carrier of R^1
sqrt (U . T2) is V11() real ext-real set
(X . T2) + (sqrt (U . T2)) is V11() real ext-real set
2 / ((X . T2) + (sqrt (U . T2))) is V11() real ext-real Element of REAL
((X . T2) + (sqrt (U . T2))) " is V11() real ext-real set
2 * (((X . T2) + (sqrt (U . T2))) ") is V11() real ext-real set
1 + (sqrt (U . T2)) is V11() real ext-real Element of REAL
2 / (1 + (sqrt (U . T2))) is V11() real ext-real Element of REAL
(1 + (sqrt (U . T2))) " is V11() real ext-real set
2 * ((1 + (sqrt (U . T2))) ") is V11() real ext-real set
T2 is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the carrier of T2 is non empty set
[: the carrier of T2, the carrier of (TOP-REAL T):] is non empty Relation-like set
bool [: the carrier of T2, the carrier of (TOP-REAL T):] is non empty set
f2 is non empty Relation-like the carrier of T2 -defined the carrier of (TOP-REAL T) -valued Function-like V26( the carrier of T2) quasi_total Element of bool [: the carrier of T2, the carrier of (TOP-REAL T):]
[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):] is non empty set
S is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of (TOP-REAL T) -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total continuous Element of bool [: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):]
dom S is non empty functional Element of bool the carrier of (TOP-REAL T)
Q is V11() real ext-real set
abs Q is V11() real ext-real non negative Element of REAL
(abs Q) * (abs Q) is V11() real ext-real non negative Element of REAL
Q * Q is V11() real ext-real set
- Q is V11() real ext-real set
- Q is V11() real ext-real set
Q is set
S is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.S.| is V11() real ext-real non negative Element of REAL
|.S.| * |.S.| is V11() real ext-real non negative Element of REAL
1 * |.S.| is V11() real ext-real non negative Element of REAL
1 - (|.S.| * |.S.|) is V11() real ext-real Element of REAL
- (|.S.| * |.S.|) is V11() real ext-real non positive set
1 + (- (|.S.| * |.S.|)) is V11() real ext-real set
1 / (1 - (|.S.| * |.S.|)) is V11() real ext-real Element of REAL
(1 - (|.S.| * |.S.|)) " is V11() real ext-real set
1 * ((1 - (|.S.| * |.S.|)) ") is V11() real ext-real set
(1 / (1 - (|.S.| * |.S.|))) * S is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.((1 / (1 - (|.S.| * |.S.|))) * S).| is V11() real ext-real non negative Element of REAL
abs (1 / (1 - (|.S.| * |.S.|))) is V11() real ext-real non negative Element of REAL
(abs (1 / (1 - (|.S.| * |.S.|)))) * |.S.| is V11() real ext-real non negative Element of REAL
|.((1 / (1 - (|.S.| * |.S.|))) * S).| * |.((1 / (1 - (|.S.| * |.S.|))) * S).| is V11() real ext-real non negative Element of REAL
(abs (1 / (1 - (|.S.| * |.S.|)))) * (abs (1 / (1 - (|.S.| * |.S.|)))) is V11() real ext-real non negative Element of REAL
((abs (1 / (1 - (|.S.| * |.S.|)))) * (abs (1 / (1 - (|.S.| * |.S.|))))) * |.S.| is V11() real ext-real non negative Element of REAL
(((abs (1 / (1 - (|.S.| * |.S.|)))) * (abs (1 / (1 - (|.S.| * |.S.|))))) * |.S.|) * |.S.| is V11() real ext-real non negative Element of REAL
(1 / (1 - (|.S.| * |.S.|))) * (1 / (1 - (|.S.| * |.S.|))) is V11() real ext-real Element of REAL
((1 / (1 - (|.S.| * |.S.|))) * (1 / (1 - (|.S.| * |.S.|)))) * |.S.| is V11() real ext-real Element of REAL
(((1 / (1 - (|.S.| * |.S.|))) * (1 / (1 - (|.S.| * |.S.|)))) * |.S.|) * |.S.| is V11() real ext-real Element of REAL
1 * 1 is non empty ordinal natural V11() real ext-real positive non negative V35() cardinal Element of REAL
(1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|)) is V11() real ext-real Element of REAL
(1 * 1) / ((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) is V11() real ext-real Element of REAL
((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) " is V11() real ext-real set
(1 * 1) * (((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) ") is V11() real ext-real set
((1 * 1) / ((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|)))) * |.S.| is V11() real ext-real Element of REAL
(((1 * 1) / ((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|)))) * |.S.|) * |.S.| is V11() real ext-real Element of REAL
(|.S.| * |.S.|) / ((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) is V11() real ext-real Element of REAL
(|.S.| * |.S.|) * (((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) ") is V11() real ext-real set
4 * |.((1 / (1 - (|.S.| * |.S.|))) * S).| is V11() real ext-real non negative Element of REAL
(4 * |.((1 / (1 - (|.S.| * |.S.|))) * S).|) * |.((1 / (1 - (|.S.| * |.S.|))) * S).| is V11() real ext-real non negative Element of REAL
1 + ((4 * |.((1 / (1 - (|.S.| * |.S.|))) * S).|) * |.((1 / (1 - (|.S.| * |.S.|))) * S).|) is non empty V11() real ext-real positive non negative Element of REAL
((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) / ((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) is V11() real ext-real Element of REAL
((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) * (((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) ") is V11() real ext-real set
4 * |.S.| is V11() real ext-real non negative Element of REAL
(4 * |.S.|) * |.S.| is V11() real ext-real non negative Element of REAL
((4 * |.S.|) * |.S.|) / ((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) is V11() real ext-real Element of REAL
((4 * |.S.|) * |.S.|) * (((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) ") is V11() real ext-real set
(((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) / ((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|)))) + (((4 * |.S.|) * |.S.|) / ((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|)))) is V11() real ext-real Element of REAL
1 + (|.S.| * |.S.|) is non empty V11() real ext-real positive non negative Element of REAL
(1 + (|.S.| * |.S.|)) * (1 + (|.S.| * |.S.|)) is non empty V11() real ext-real positive non negative Element of REAL
((1 + (|.S.| * |.S.|)) * (1 + (|.S.| * |.S.|))) / ((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) is V11() real ext-real Element of REAL
((1 + (|.S.| * |.S.|)) * (1 + (|.S.| * |.S.|))) * (((1 - (|.S.| * |.S.|)) * (1 - (|.S.| * |.S.|))) ") is V11() real ext-real set
(1 + (|.S.| * |.S.|)) / (1 - (|.S.| * |.S.|)) is V11() real ext-real Element of REAL
(1 + (|.S.| * |.S.|)) * ((1 - (|.S.| * |.S.|)) ") is V11() real ext-real set
((1 + (|.S.| * |.S.|)) / (1 - (|.S.| * |.S.|))) ^2 is V11() real ext-real Element of REAL
((1 + (|.S.| * |.S.|)) / (1 - (|.S.| * |.S.|))) * ((1 + (|.S.| * |.S.|)) / (1 - (|.S.| * |.S.|))) is V11() real ext-real set
sqrt (1 + ((4 * |.((1 / (1 - (|.S.| * |.S.|))) * S).|) * |.((1 / (1 - (|.S.| * |.S.|))) * S).|)) is V11() real ext-real Element of REAL
1 + (sqrt (1 + ((4 * |.((1 / (1 - (|.S.| * |.S.|))) * S).|) * |.((1 / (1 - (|.S.| * |.S.|))) * S).|))) is V11() real ext-real Element of REAL
(1 - (|.S.| * |.S.|)) / (1 - (|.S.| * |.S.|)) is V11() real ext-real Element of REAL
(1 - (|.S.| * |.S.|)) * ((1 - (|.S.| * |.S.|)) ") is V11() real ext-real set
((1 - (|.S.| * |.S.|)) / (1 - (|.S.| * |.S.|))) + ((1 + (|.S.| * |.S.|)) / (1 - (|.S.| * |.S.|))) is V11() real ext-real Element of REAL
2 / (1 - (|.S.| * |.S.|)) is V11() real ext-real Element of REAL
2 * ((1 - (|.S.| * |.S.|)) ") is V11() real ext-real set
U5 is set
y is set
S . y is Relation-like Function-like set
M2 . ((1 / (1 - (|.S.| * |.S.|))) * S) is V11() real ext-real set
(M2 . ((1 / (1 - (|.S.| * |.S.|))) * S)) * ((1 / (1 - (|.S.| * |.S.|))) * S) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
2 / (2 / (1 - (|.S.| * |.S.|))) is V11() real ext-real Element of REAL
(2 / (1 - (|.S.| * |.S.|))) " is V11() real ext-real set
2 * ((2 / (1 - (|.S.| * |.S.|))) ") is V11() real ext-real set
(2 / (2 / (1 - (|.S.| * |.S.|)))) * ((1 / (1 - (|.S.| * |.S.|))) * S) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
(1 - (|.S.| * |.S.|)) * ((1 / (1 - (|.S.| * |.S.|))) * S) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
((1 - (|.S.| * |.S.|)) / (1 - (|.S.| * |.S.|))) * S is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
1 * S is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
S is set
S . S is Relation-like Function-like set
rng S is non empty functional Element of bool the carrier of (TOP-REAL T)
U4 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
M2 . U4 is V11() real ext-real set
(M2 . U4) * U4 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.U4.| is V11() real ext-real non negative Element of REAL
4 * |.U4.| is V11() real ext-real non negative Element of REAL
(4 * |.U4.|) * |.U4.| is V11() real ext-real non negative Element of REAL
1 + ((4 * |.U4.|) * |.U4.|) is non empty V11() real ext-real positive non negative Element of REAL
sqrt (1 + ((4 * |.U4.|) * |.U4.|)) is V11() real ext-real Element of REAL
1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))) is V11() real ext-real Element of REAL
2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) is V11() real ext-real Element of REAL
(1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) " is V11() real ext-real set
2 * ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ") is V11() real ext-real set
(2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) * U4 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
U5 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.U5.| is V11() real ext-real non negative Element of REAL
abs (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) is V11() real ext-real non negative Element of REAL
(abs (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))))) * |.U4.| is V11() real ext-real non negative Element of REAL
|.U5.| * |.U5.| is V11() real ext-real non negative Element of REAL
(abs (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))))) * (abs (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))))) is V11() real ext-real non negative Element of REAL
((abs (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))))) * (abs (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))))) * |.U4.| is V11() real ext-real non negative Element of REAL
(((abs (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))))) * (abs (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))))) * |.U4.|) * |.U4.| is V11() real ext-real non negative Element of REAL
(2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) * (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) is V11() real ext-real Element of REAL
((2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) * (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))))) * |.U4.| is V11() real ext-real Element of REAL
(((2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) * (2 / (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))))) * |.U4.|) * |.U4.| is V11() real ext-real Element of REAL
2 * 2 is non empty ordinal natural V11() real ext-real positive non negative V35() cardinal Element of REAL
(1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) * (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) is V11() real ext-real Element of REAL
(2 * 2) / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) * (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) is V11() real ext-real Element of REAL
((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) * (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) " is V11() real ext-real set
(2 * 2) * (((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) * (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) ") is V11() real ext-real set
((2 * 2) / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) * (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))))) * |.U4.| is V11() real ext-real Element of REAL
(((2 * 2) / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) * (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))))) * |.U4.|) * |.U4.| is V11() real ext-real Element of REAL
(1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2 is V11() real ext-real Element of REAL
(1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) * (1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) is V11() real ext-real set
1 * ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) is V11() real ext-real Element of REAL
4 / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) is V11() real ext-real Element of REAL
((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) " is V11() real ext-real set
4 * (((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) ") is V11() real ext-real set
(4 / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2)) * |.U4.| is V11() real ext-real Element of REAL
((4 / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2)) * |.U4.|) * |.U4.| is V11() real ext-real Element of REAL
(((4 / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2)) * |.U4.|) * |.U4.|) * ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) is V11() real ext-real Element of REAL
((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) is V11() real ext-real Element of REAL
((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) * (((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) ") is V11() real ext-real set
(((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2)) * 4 is V11() real ext-real Element of REAL
((((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2)) * 4) * |.U4.| is V11() real ext-real Element of REAL
(((((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2) / ((1 + (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) ^2)) * 4) * |.U4.|) * |.U4.| is V11() real ext-real Element of REAL
1 * 4 is non empty ordinal natural V11() real ext-real positive non negative V35() cardinal Element of REAL
(1 * 4) * |.U4.| is V11() real ext-real non negative Element of REAL
((1 * 4) * |.U4.|) * |.U4.| is V11() real ext-real non negative Element of REAL
2 * (sqrt (1 + ((4 * |.U4.|) * |.U4.|))) is V11() real ext-real Element of REAL
1 + (2 * (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) is V11() real ext-real Element of REAL
(sqrt (1 + ((4 * |.U4.|) * |.U4.|))) ^2 is V11() real ext-real Element of REAL
(sqrt (1 + ((4 * |.U4.|) * |.U4.|))) * (sqrt (1 + ((4 * |.U4.|) * |.U4.|))) is V11() real ext-real set
(1 + (2 * (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) + ((sqrt (1 + ((4 * |.U4.|) * |.U4.|))) ^2) is V11() real ext-real Element of REAL
(1 + (2 * (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) + (1 + ((4 * |.U4.|) * |.U4.|)) is V11() real ext-real Element of REAL
2 + (2 * (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) is V11() real ext-real Element of REAL
(2 + (2 * (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) + ((4 * |.U4.|) * |.U4.|) is V11() real ext-real Element of REAL
((2 + (2 * (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) + ((4 * |.U4.|) * |.U4.|)) - ((4 * |.U4.|) * |.U4.|) is V11() real ext-real Element of REAL
- ((4 * |.U4.|) * |.U4.|) is V11() real ext-real non positive set
((2 + (2 * (sqrt (1 + ((4 * |.U4.|) * |.U4.|))))) + ((4 * |.U4.|) * |.U4.|)) + (- ((4 * |.U4.|) * |.U4.|)) is V11() real ext-real set
((4 * |.U4.|) * |.U4.|) - ((4 * |.U4.|) * |.U4.|) is V11() real ext-real Element of REAL
((4 * |.U4.|) * |.U4.|) + (- ((4 * |.U4.|) * |.U4.|)) is V11() real ext-real set
{} / 2 is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of REAL
{} * (2 ") is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() set
(2 * (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) / 2 is V11() real ext-real Element of REAL
(2 * (sqrt (1 + ((4 * |.U4.|) * |.U4.|)))) * (2 ") is V11() real ext-real set
|.U5.| ^2 is V11() real ext-real Element of REAL
|.U5.| * |.U5.| is V11() real ext-real non negative set
0. (TOP-REAL M) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V54( TOP-REAL M) V124() V125() V126() Element of the carrier of (TOP-REAL M)
the ZeroF of (TOP-REAL M) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
- (0. (TOP-REAL M)) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
U5 + (- (0. (TOP-REAL M))) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
the U7 of (TOP-REAL M) is non empty Relation-like [: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):] -defined the carrier of (TOP-REAL M) -valued Function-like V26([: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):]
[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL M), the carrier of (TOP-REAL M):], the carrier of (TOP-REAL M):] is non empty set
K155( the carrier of (TOP-REAL M), the U7 of (TOP-REAL M),U5,(- (0. (TOP-REAL M)))) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.(U5 + (- (0. (TOP-REAL M)))).| is V11() real ext-real non negative Element of REAL
|.(- (0. (TOP-REAL M))).| is V11() real ext-real non negative Element of REAL
|.U5.| + |.(- (0. (TOP-REAL M))).| is V11() real ext-real non negative Element of REAL
|.(0. (TOP-REAL M)).| is V11() real ext-real non negative Element of REAL
|.U5.| + |.(0. (TOP-REAL M)).| is V11() real ext-real non negative Element of REAL
|.U5.| + {} is V11() real ext-real non negative Element of REAL
U5 - (0. (TOP-REAL M)) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
K184((TOP-REAL M),U5,(- (0. (TOP-REAL M)))) is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.(U5 - (0. (TOP-REAL M))).| is V11() real ext-real non negative Element of REAL
Ball ((0. (TOP-REAL M)),1) is non empty functional open V162( TOP-REAL M) Element of bool the carrier of (TOP-REAL M)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M) : not 1 <= |.(b₁ - (0. (TOP-REAL M))).| } is set
[#] (T,(0. (TOP-REAL T)),1) is non empty non proper open closed dense non boundary Element of bool the carrier of (T,(0. (TOP-REAL T)),1)
the carrier of (T,(0. (TOP-REAL T)),1) is non empty set
bool the carrier of (T,(0. (TOP-REAL T)),1) is non empty set
rng S is non empty functional Element of bool the carrier of (TOP-REAL T)
[: the carrier of (TOP-REAL T), the carrier of (T):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL T), the carrier of (T):] is non empty set
Q is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of (T) -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total Element of bool [: the carrier of (TOP-REAL T), the carrier of (T):]
dom Q is non empty functional Element of bool the carrier of (TOP-REAL T)
c₂ " is non empty Relation-like the carrier of (TOP-REAL T) -defined the carrier of (T) -valued Function-like V26( the carrier of (TOP-REAL T)) quasi_total Element of bool [: the carrier of (TOP-REAL T), the carrier of (T):]
dom (c₂ ") is non empty functional Element of bool the carrier of (TOP-REAL T)
S is set
Q . S is set
(c₂ ") . S is set
U4 is Relation-like Function-like set
U4 " is Relation-like Function-like set
rng Q is non empty Element of bool the carrier of (T)
U5 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
Q . U5 is set
M2 . U5 is V11() real ext-real set
(M2 . U5) * U5 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.U5.| is V11() real ext-real non negative Element of REAL
4 * |.U5.| is V11() real ext-real non negative Element of REAL
(4 * |.U5.|) * |.U5.| is V11() real ext-real non negative Element of REAL
1 + ((4 * |.U5.|) * |.U5.|) is non empty V11() real ext-real positive non negative Element of REAL
sqrt (1 + ((4 * |.U5.|) * |.U5.|)) is V11() real ext-real Element of REAL
1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))) is V11() real ext-real Element of REAL
2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) is V11() real ext-real Element of REAL
(1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) " is V11() real ext-real set
2 * ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ") is V11() real ext-real set
(2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) * U5 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
y is Element of the carrier of (T)
q is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
|.q.| is V11() real ext-real non negative Element of REAL
abs (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) is V11() real ext-real non negative Element of REAL
(abs (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * |.U5.| is V11() real ext-real non negative Element of REAL
|.q.| * |.q.| is V11() real ext-real non negative Element of REAL
(abs (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * (abs (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) is V11() real ext-real non negative Element of REAL
((abs (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * (abs (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))))) * |.U5.| is V11() real ext-real non negative Element of REAL
(((abs (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * (abs (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))))) * |.U5.|) * |.U5.| is V11() real ext-real non negative Element of REAL
(2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) * (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) is V11() real ext-real Element of REAL
((2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) * (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * |.U5.| is V11() real ext-real Element of REAL
(((2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) * (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * |.U5.|) * |.U5.| is V11() real ext-real Element of REAL
2 * 2 is non empty ordinal natural V11() real ext-real positive non negative V35() cardinal Element of REAL
(1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) is V11() real ext-real Element of REAL
(2 * 2) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) is V11() real ext-real Element of REAL
((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) " is V11() real ext-real set
(2 * 2) * (((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) ") is V11() real ext-real set
((2 * 2) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * |.U5.| is V11() real ext-real Element of REAL
(((2 * 2) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * |.U5.|) * |.U5.| is V11() real ext-real Element of REAL
(1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2 is V11() real ext-real Element of REAL
(1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) is V11() real ext-real set
((4 * |.U5.|) * |.U5.|) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) is V11() real ext-real Element of REAL
((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) " is V11() real ext-real set
((4 * |.U5.|) * |.U5.|) * (((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) ") is V11() real ext-real set
1 - (|.q.| * |.q.|) is V11() real ext-real Element of REAL
- (|.q.| * |.q.|) is V11() real ext-real non positive set
1 + (- (|.q.| * |.q.|)) is V11() real ext-real set
((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) is V11() real ext-real Element of REAL
((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) * (((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) ") is V11() real ext-real set
(((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2)) - (((4 * |.U5.|) * |.U5.|) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2)) is V11() real ext-real Element of REAL
- (((4 * |.U5.|) * |.U5.|) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2)) is V11() real ext-real set
(((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2)) + (- (((4 * |.U5.|) * |.U5.|) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2))) is V11() real ext-real set
2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))) is V11() real ext-real Element of REAL
1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) is V11() real ext-real Element of REAL
(sqrt (1 + ((4 * |.U5.|) * |.U5.|))) ^2 is V11() real ext-real Element of REAL
(sqrt (1 + ((4 * |.U5.|) * |.U5.|))) * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))) is V11() real ext-real set
(1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) + ((sqrt (1 + ((4 * |.U5.|) * |.U5.|))) ^2) is V11() real ext-real Element of REAL
((1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) + ((sqrt (1 + ((4 * |.U5.|) * |.U5.|))) ^2)) - ((4 * |.U5.|) * |.U5.|) is V11() real ext-real Element of REAL
- ((4 * |.U5.|) * |.U5.|) is V11() real ext-real non positive set
((1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) + ((sqrt (1 + ((4 * |.U5.|) * |.U5.|))) ^2)) + (- ((4 * |.U5.|) * |.U5.|)) is V11() real ext-real set
(((1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) + ((sqrt (1 + ((4 * |.U5.|) * |.U5.|))) ^2)) - ((4 * |.U5.|) * |.U5.|)) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) is V11() real ext-real Element of REAL
(((1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) + ((sqrt (1 + ((4 * |.U5.|) * |.U5.|))) ^2)) - ((4 * |.U5.|) * |.U5.|)) * (((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) ") is V11() real ext-real set
(1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) + (1 + ((4 * |.U5.|) * |.U5.|)) is V11() real ext-real Element of REAL
((1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) + (1 + ((4 * |.U5.|) * |.U5.|))) - ((4 * |.U5.|) * |.U5.|) is V11() real ext-real Element of REAL
((1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) + (1 + ((4 * |.U5.|) * |.U5.|))) + (- ((4 * |.U5.|) * |.U5.|)) is V11() real ext-real set
(((1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) + (1 + ((4 * |.U5.|) * |.U5.|))) - ((4 * |.U5.|) * |.U5.|)) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) is V11() real ext-real Element of REAL
(((1 + (2 * (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) + (1 + ((4 * |.U5.|) * |.U5.|))) - ((4 * |.U5.|) * |.U5.|)) * (((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) ^2) ") is V11() real ext-real set
2 * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) is V11() real ext-real Element of REAL
(2 * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) / ((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) is V11() real ext-real Element of REAL
(2 * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) * (((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) ") is V11() real ext-real set
c₂ . (Q . S) is Relation-like Function-like set
1 / (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) is V11() real ext-real Element of REAL
(2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) " is V11() real ext-real set
1 * ((2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) ") is V11() real ext-real set
(1 / (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * q is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
(1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) / 2 is V11() real ext-real Element of REAL
(1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) * (2 ") is V11() real ext-real set
((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) / 2) * q is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) / 2) * (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) is V11() real ext-real Element of REAL
(((1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))) / 2) * (2 / (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * U5 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
(2 * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) / (2 * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) is V11() real ext-real Element of REAL
(2 * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) " is V11() real ext-real set
(2 * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) * ((2 * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) ") is V11() real ext-real set
((2 * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|))))) / (2 * (1 + (sqrt (1 + ((4 * |.U5.|) * |.U5.|)))))) * U5 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
1 * U5 is Relation-like NAT -defined Function-like V35() M -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL M)
[(Q . S),S] is set
{(Q . S),S} is non empty set
{(Q . S)} is non empty trivial 1 -element set
{{(Q . S),S},{(Q . S)}} is non empty set
[S,(Q . S)] is set
{S,(Q . S)} is non empty set
{S} is non empty trivial 1 -element set
{{S,(Q . S)},{S}} is non empty set
U4 ~ is Relation-like set
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
(T) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
0. (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V54( TOP-REAL T) V124() V125() V126() Element of the carrier of (TOP-REAL T)
the ZeroF of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(T,(0. (TOP-REAL T)),1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball ((0. (TOP-REAL T)),1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not 1 <= |.(b₁ - (0. (TOP-REAL T))).| } is set
(TOP-REAL T) | (Ball ((0. (TOP-REAL T)),1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the carrier of (T) is non empty set
c₂ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
M is non empty V11() real ext-real positive non negative set
(T,c₂,M) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball (c₂,M) is non empty functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not M <= |.(b₁ - c₂).| } is set
(TOP-REAL T) | (Ball (c₂,M)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the carrier of (T,c₂,M) is non empty set
[: the carrier of (T), the carrier of (T,c₂,M):] is non empty Relation-like set
bool [: the carrier of (T), the carrier of (T,c₂,M):] is non empty set
X is non empty Relation-like the carrier of (T) -defined the carrier of (T,c₂,M) -valued Function-like V26( the carrier of (T)) quasi_total Element of bool [: the carrier of (T), the carrier of (T,c₂,M):]
X1 is non empty ordinal natural V11() real ext-real positive non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of NAT
TOP-REAL X1 is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable L15()
the carrier of (TOP-REAL X1) is non empty functional set
p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(X1,p,M) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL X1
Ball (p,M) is non empty functional open V162( TOP-REAL X1) Element of bool the carrier of (TOP-REAL X1)
bool the carrier of (TOP-REAL X1) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1) : not M <= |.(b₁ - p).| } is set
(TOP-REAL X1) | (Ball (p,M)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL X1
S1 is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
M * S1 is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(M * S1) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
the U7 of (TOP-REAL X1) is non empty Relation-like [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):] -defined the carrier of (TOP-REAL X1) -valued Function-like V26([: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):]
[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty set
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(M * S1),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):] is non empty set
S1 is non empty Relation-like the carrier of (TOP-REAL X1) -defined the carrier of (TOP-REAL X1) -valued Function-like V26( the carrier of (TOP-REAL X1)) quasi_total Element of bool [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]
1 / M is non empty V11() real ext-real positive non negative Element of REAL
M " is non empty V11() real ext-real positive non negative set
1 * (M ") is non empty V11() real ext-real positive non negative set
U2 is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
U2 - p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
- p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K184((TOP-REAL X1),U2,(- p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
the U7 of (TOP-REAL X1) is non empty Relation-like [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):] -defined the carrier of (TOP-REAL X1) -valued Function-like V26([: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):]
[:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty set
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),U2,(- p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(1 / M) * (U2 - p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
U2 is non empty Relation-like the carrier of (TOP-REAL X1) -defined the carrier of (TOP-REAL X1) -valued Function-like V26( the carrier of (TOP-REAL X1)) quasi_total Element of bool [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]
(TOP-REAL X1) --> p is non empty Relation-like the carrier of (TOP-REAL X1) -defined the carrier of (TOP-REAL X1) -valued Function-like V26( the carrier of (TOP-REAL X1)) quasi_total continuous Element of bool [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]
K445( the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1),p) is non empty Relation-like the carrier of (TOP-REAL X1) -defined the carrier of (TOP-REAL X1) -valued Function-like V26( the carrier of (TOP-REAL X1)) quasi_total Element of bool [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]
id (TOP-REAL X1) is non empty Relation-like the carrier of (TOP-REAL X1) -defined the carrier of (TOP-REAL X1) -valued Function-like V26( the carrier of (TOP-REAL X1)) quasi_total continuous Element of bool [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]
K66( the carrier of (TOP-REAL X1)) is non empty Relation-like the carrier of (TOP-REAL X1) -defined the carrier of (TOP-REAL X1) -valued Function-like one-to-one V26( the carrier of (TOP-REAL X1)) quasi_total Element of bool [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]
dom U2 is non empty functional Element of bool the carrier of (TOP-REAL X1)
U2 | (Ball (p,M)) is Relation-like the carrier of (TOP-REAL X1) -defined Ball (p,M) -defined the carrier of (TOP-REAL X1) -defined the carrier of (TOP-REAL X1) -valued Function-like Element of bool [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]
dom (U2 | (Ball (p,M))) is functional Element of bool (Ball (p,M))
bool (Ball (p,M)) is non empty set
- (1 / M) is non empty V11() real ext-real non positive negative Element of REAL
U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
U2 . U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(id (TOP-REAL X1)) . U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(1 / M) * ((id (TOP-REAL X1)) . U) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
((TOP-REAL X1) --> p) . U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(- (1 / M)) * (((TOP-REAL X1) --> p) . U) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
((1 / M) * ((id (TOP-REAL X1)) . U)) + ((- (1 / M)) * (((TOP-REAL X1) --> p) . U)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
the U7 of (TOP-REAL X1) is non empty Relation-like [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):] -defined the carrier of (TOP-REAL X1) -valued Function-like V26([: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):]
[:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty set
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),((1 / M) * ((id (TOP-REAL X1)) . U)),((- (1 / M)) * (((TOP-REAL X1) --> p) . U))) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(1 / M) * U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
((1 / M) * U) + ((- (1 / M)) * (((TOP-REAL X1) --> p) . U)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),((1 / M) * U),((- (1 / M)) * (((TOP-REAL X1) --> p) . U))) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(- (1 / M)) * p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
((1 / M) * U) + ((- (1 / M)) * p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),((1 / M) * U),((- (1 / M)) * p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(1 / M) * p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
((1 / M) * U) - ((1 / M) * p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
- ((1 / M) * p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K184((TOP-REAL X1),((1 / M) * U),(- ((1 / M) * p))) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),((1 / M) * U),(- ((1 / M) * p))) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
U - p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
- p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K184((TOP-REAL X1),U,(- p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),U,(- p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(1 / M) * (U - p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
dom X is non empty Element of bool the carrier of (T)
bool the carrier of (T) is non empty set
[#] (T) is non empty non proper open closed dense non boundary Element of bool the carrier of (T)
U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
S1 . U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(id (TOP-REAL X1)) . U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
M * ((id (TOP-REAL X1)) . U) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
((TOP-REAL X1) --> p) . U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
1 * (((TOP-REAL X1) --> p) . U) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(M * ((id (TOP-REAL X1)) . U)) + (1 * (((TOP-REAL X1) --> p) . U)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
the U7 of (TOP-REAL X1) is non empty Relation-like [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):] -defined the carrier of (TOP-REAL X1) -valued Function-like V26([: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):]
[:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty set
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(M * ((id (TOP-REAL X1)) . U)),(1 * (((TOP-REAL X1) --> p) . U))) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(M * ((id (TOP-REAL X1)) . U)) + (((TOP-REAL X1) --> p) . U) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(M * ((id (TOP-REAL X1)) . U)),(((TOP-REAL X1) --> p) . U)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
M * U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(M * U) + (((TOP-REAL X1) --> p) . U) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(M * U),(((TOP-REAL X1) --> p) . U)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(M * U) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(M * U),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
the carrier of (X1,p,M) is non empty set
S1 | (Ball ((0. (TOP-REAL T)),1)) is Relation-like Ball ((0. (TOP-REAL T)),1) -defined the carrier of (TOP-REAL X1) -defined the carrier of (TOP-REAL X1) -valued Function-like Element of bool [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]
U is set
X . U is set
(S1 | (Ball ((0. (TOP-REAL T)),1))) . U is Relation-like Function-like set
f is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
M * f is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(M * f) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
the U7 of (TOP-REAL X1) is non empty Relation-like [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):] -defined the carrier of (TOP-REAL X1) -valued Function-like V26([: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):]
[:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty set
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(M * f),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
S1 . f is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(1 / M) * M is non empty V11() real ext-real positive non negative Element of REAL
rng X is non empty Element of bool the carrier of (T,c₂,M)
bool the carrier of (T,c₂,M) is non empty set
[#] (X1,p,M) is non empty non proper open closed dense non boundary Element of bool the carrier of (X1,p,M)
bool the carrier of (X1,p,M) is non empty set
U is set
f is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
f - p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
- p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K184((TOP-REAL X1),f,(- p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
the U7 of (TOP-REAL X1) is non empty Relation-like [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):] -defined the carrier of (TOP-REAL X1) -valued Function-like V26([: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):]
[:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty set
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),f,(- p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
U is V11() real ext-real Element of REAL
U * (f - p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
0. (TOP-REAL X1) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V54( TOP-REAL X1) V124() V125() V126() Element of the carrier of (TOP-REAL X1)
the ZeroF of (TOP-REAL X1) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(U * (f - p)) - (0. (TOP-REAL X1)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
- (0. (TOP-REAL X1)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K184((TOP-REAL X1),(U * (f - p)),(- (0. (TOP-REAL X1)))) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(U * (f - p)),(- (0. (TOP-REAL X1)))) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
|.((U * (f - p)) - (0. (TOP-REAL X1))).| is V11() real ext-real non negative Element of REAL
|.(U * (f - p)).| is V11() real ext-real non negative Element of REAL
abs U is V11() real ext-real non negative Element of REAL
|.(f - p).| is V11() real ext-real non negative Element of REAL
(abs U) * |.(f - p).| is V11() real ext-real non negative Element of REAL
U * |.(f - p).| is V11() real ext-real Element of REAL
(1 / M) * |.(f - p).| is V11() real ext-real non negative Element of REAL
X . (U * (f - p)) is set
M * (U * (f - p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(M * (U * (f - p))) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(M * (U * (f - p))),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
M * (1 / M) is non empty V11() real ext-real positive non negative Element of REAL
(M * (1 / M)) * (f - p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
((M * (1 / M)) * (f - p)) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),((M * (1 / M)) * (f - p)),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(f - p) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(f - p),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
U is set
f is set
X . U is set
X . f is set
U3 is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
X . U3 is set
M * U3 is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(M * U3) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
the U7 of (TOP-REAL X1) is non empty Relation-like [: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):] -defined the carrier of (TOP-REAL X1) -valued Function-like V26([: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):]
[:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL X1), the carrier of (TOP-REAL X1):], the carrier of (TOP-REAL X1):] is non empty set
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(M * U3),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
X . U is set
M * U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(M * U) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(M * U),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
((M * U3) + p) - p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
- p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K184((TOP-REAL X1),((M * U3) + p),(- p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),((M * U3) + p),(- p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
X " is non empty Relation-like the carrier of (T,c₂,M) -defined the carrier of (T) -valued Function-like V26( the carrier of (T,c₂,M)) quasi_total Element of bool [: the carrier of (T,c₂,M), the carrier of (T):]
[: the carrier of (T,c₂,M), the carrier of (T):] is non empty Relation-like set
bool [: the carrier of (T,c₂,M), the carrier of (T):] is non empty set
dom (X ") is non empty Element of bool the carrier of (T,c₂,M)
f is set
(X ") . f is set
(U2 | (Ball (p,M))) . f is Relation-like Function-like set
U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
U - p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K184((TOP-REAL X1),U,(- p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),U,(- p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(1 / M) * (U - p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
0. (TOP-REAL X1) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V54( TOP-REAL X1) V124() V125() V126() Element of the carrier of (TOP-REAL X1)
the ZeroF of (TOP-REAL X1) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
((1 / M) * (U - p)) - (0. (TOP-REAL X1)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
- (0. (TOP-REAL X1)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K184((TOP-REAL X1),((1 / M) * (U - p)),(- (0. (TOP-REAL X1)))) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),((1 / M) * (U - p)),(- (0. (TOP-REAL X1)))) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
|.(((1 / M) * (U - p)) - (0. (TOP-REAL X1))).| is V11() real ext-real non negative Element of REAL
|.((1 / M) * (U - p)).| is V11() real ext-real non negative Element of REAL
U3 is V11() real ext-real Element of REAL
abs U3 is V11() real ext-real non negative Element of REAL
|.(U - p).| is V11() real ext-real non negative Element of REAL
(abs U3) * |.(U - p).| is V11() real ext-real non negative Element of REAL
U3 * |.(U - p).| is V11() real ext-real Element of REAL
(1 / M) * |.(U - p).| is V11() real ext-real non negative Element of REAL
X . ((1 / M) * (U - p)) is set
M * ((1 / M) * (U - p)) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(M * ((1 / M) * (U - p))) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(M * ((1 / M) * (U - p))),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
M * (1 / M) is non empty V11() real ext-real positive non negative Element of REAL
(M * (1 / M)) * (U - p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
((M * (1 / M)) * (U - p)) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),((M * (1 / M)) * (U - p)),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
(U - p) + p is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
K155( the carrier of (TOP-REAL X1), the U7 of (TOP-REAL X1),(U - p),p) is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
U is Relation-like Function-like set
U " is Relation-like Function-like set
(U ") . f is set
U2 . U is Relation-like NAT -defined Function-like V35() X1 -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL X1)
dom S1 is non empty functional Element of bool the carrier of (TOP-REAL X1)
dom (S1 | (Ball ((0. (TOP-REAL T)),1))) is functional Element of bool (Ball ((0. (TOP-REAL T)),1))
bool (Ball ((0. (TOP-REAL T)),1)) is non empty set
T is ordinal natural V11() real ext-real non negative V35() cardinal set
(T) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
0. (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V54( TOP-REAL T) V124() V125() V126() Element of the carrier of (TOP-REAL T)
the carrier of (TOP-REAL T) is non empty functional set
the ZeroF of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(T,(0. (TOP-REAL T)),1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball ((0. (TOP-REAL T)),1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not 1 <= |.(b₁ - (0. (TOP-REAL T))).| } is set
(TOP-REAL T) | (Ball ((0. (TOP-REAL T)),1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the carrier of (T) is non empty set
c₂ is ordinal natural V11() real ext-real non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of NAT
TOP-REAL c₂ is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable L15()
the carrier of (TOP-REAL c₂) is non empty functional set
X1 is Element of the carrier of (T)
p is Relation-like NAT -defined Function-like V35() c₂ -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL c₂)
|.p.| is V11() real ext-real non negative Element of REAL
|.p.| * |.p.| is V11() real ext-real non negative Element of REAL
1 - (|.p.| * |.p.|) is V11() real ext-real Element of REAL
- (|.p.| * |.p.|) is V11() real ext-real non positive set
1 + (- (|.p.| * |.p.|)) is V11() real ext-real set
1 / (1 - (|.p.| * |.p.|)) is V11() real ext-real Element of REAL
(1 - (|.p.| * |.p.|)) " is V11() real ext-real set
1 * ((1 - (|.p.| * |.p.|)) ") is V11() real ext-real set
(1 / (1 - (|.p.| * |.p.|))) * p is Relation-like NAT -defined Function-like V35() c₂ -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL c₂)
U1 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
[: the carrier of (T), the carrier of (TOP-REAL T):] is non empty Relation-like set
bool [: the carrier of (T), the carrier of (TOP-REAL T):] is non empty set
X1 is non empty Relation-like the carrier of (T) -defined the carrier of (TOP-REAL T) -valued Function-like V26( the carrier of (T)) quasi_total Element of bool [: the carrier of (T), the carrier of (TOP-REAL T):]
p is Element of the carrier of (T)
X1 . p is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
p1 is Relation-like NAT -defined Function-like V35() c₂ -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL c₂)
|.p1.| is V11() real ext-real non negative Element of REAL
|.p1.| * |.p1.| is V11() real ext-real non negative Element of REAL
1 - (|.p1.| * |.p1.|) is V11() real ext-real Element of REAL
- (|.p1.| * |.p1.|) is V11() real ext-real non positive set
1 + (- (|.p1.| * |.p1.|)) is V11() real ext-real set
1 / (1 - (|.p1.| * |.p1.|)) is V11() real ext-real Element of REAL
(1 - (|.p1.| * |.p1.|)) " is V11() real ext-real set
1 * ((1 - (|.p1.| * |.p1.|)) ") is V11() real ext-real set
(1 / (1 - (|.p1.| * |.p1.|))) * p1 is Relation-like NAT -defined Function-like V35() c₂ -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL c₂)
U1 is Relation-like NAT -defined Function-like V35() c₂ -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL c₂)
|.U1.| is V11() real ext-real non negative Element of REAL
|.U1.| * |.U1.| is V11() real ext-real non negative Element of REAL
1 - (|.U1.| * |.U1.|) is V11() real ext-real Element of REAL
- (|.U1.| * |.U1.|) is V11() real ext-real non positive set
1 + (- (|.U1.| * |.U1.|)) is V11() real ext-real set
1 / (1 - (|.U1.| * |.U1.|)) is V11() real ext-real Element of REAL
(1 - (|.U1.| * |.U1.|)) " is V11() real ext-real set
1 * ((1 - (|.U1.| * |.U1.|)) ") is V11() real ext-real set
(1 / (1 - (|.U1.| * |.U1.|))) * U1 is Relation-like NAT -defined Function-like V35() c₂ -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL c₂)
p is non empty Relation-like the carrier of (T) -defined the carrier of (TOP-REAL T) -valued Function-like V26( the carrier of (T)) quasi_total Element of bool [: the carrier of (T), the carrier of (TOP-REAL T):]
TOP-REAL {} is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
0. (TOP-REAL {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V54( TOP-REAL {}) V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
the carrier of (TOP-REAL {}) is non empty functional set
the ZeroF of (TOP-REAL {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
Ball ((0. (TOP-REAL {})),1) is non empty functional open Element of bool the carrier of (TOP-REAL {})
bool the carrier of (TOP-REAL {}) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() {} -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL {}) : not 1 <= |.(b₁ - (0. (TOP-REAL {}))).| } is set
X1 is set
p is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
p - (0. (TOP-REAL {})) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
- (0. (TOP-REAL {})) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
K184((TOP-REAL {}),p,(- (0. (TOP-REAL {})))) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
the U7 of (TOP-REAL {}) is non empty Relation-like [: the carrier of (TOP-REAL {}), the carrier of (TOP-REAL {}):] -defined the carrier of (TOP-REAL {}) -valued Function-like V26([: the carrier of (TOP-REAL {}), the carrier of (TOP-REAL {}):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL {}), the carrier of (TOP-REAL {}):], the carrier of (TOP-REAL {}):]
[: the carrier of (TOP-REAL {}), the carrier of (TOP-REAL {}):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL {}), the carrier of (TOP-REAL {}):], the carrier of (TOP-REAL {}):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL {}), the carrier of (TOP-REAL {}):], the carrier of (TOP-REAL {}):] is non empty set
K155( the carrier of (TOP-REAL {}), the U7 of (TOP-REAL {}),p,(- (0. (TOP-REAL {})))) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
|.(p - (0. (TOP-REAL {}))).| is V11() real ext-real non negative Element of REAL
[#] (TOP-REAL {}) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL {})
{(0. (TOP-REAL {}))} is non empty trivial functional 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
c₂ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
M is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
X is non empty V11() real ext-real positive non negative set
(T,c₂,X) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball (c₂,X) is non empty functional open Element of bool the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not X <= |.(b₁ - c₂).| } is set
(TOP-REAL T) | (Ball (c₂,X)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
X1 is non empty V11() real ext-real positive non negative set
(T,M,X1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball (M,X1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not X1 <= |.(b₁ - M).| } is set
(TOP-REAL T) | (Ball (M,X1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
p is ordinal natural V11() real ext-real non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of NAT
TOP-REAL p is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable L15()
the carrier of (TOP-REAL p) is non empty functional set
p1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
TOP-REAL {} is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
0. (TOP-REAL {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V54( TOP-REAL {}) V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
the carrier of (TOP-REAL {}) is non empty functional set
the ZeroF of (TOP-REAL {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
U1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
S2 is set
X2 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
X2 - c₂ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
- c₂ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
K184((TOP-REAL T),X2,(- c₂)) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
the U7 of (TOP-REAL T) is non empty Relation-like [: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):] -defined the carrier of (TOP-REAL T) -valued Function-like V26([: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):], the carrier of (TOP-REAL T):]
[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):], the carrier of (TOP-REAL T):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL T), the carrier of (TOP-REAL T):], the carrier of (TOP-REAL T):] is non empty set
K155( the carrier of (TOP-REAL T), the U7 of (TOP-REAL T),X2,(- c₂)) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
|.(X2 - c₂).| is V11() real ext-real non negative Element of REAL
U is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
U - p1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
- p1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
K184((TOP-REAL p),U,(- p1)) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
the U7 of (TOP-REAL p) is non empty Relation-like [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] -defined the carrier of (TOP-REAL p) -valued Function-like V26([: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):]
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):] is non empty set
K155( the carrier of (TOP-REAL p), the U7 of (TOP-REAL p),U,(- p1)) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
|.(U - p1).| is V11() real ext-real non negative Element of REAL
X2 is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
X2 - M is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
- M is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
K184((TOP-REAL T),X2,(- M)) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
K155( the carrier of (TOP-REAL T), the U7 of (TOP-REAL T),X2,(- M)) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
|.(X2 - M).| is V11() real ext-real non negative Element of REAL
U is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
U - U1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
- U1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
K184((TOP-REAL p),U,(- U1)) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
K155( the carrier of (TOP-REAL p), the U7 of (TOP-REAL p),U,(- U1)) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
|.(U - U1).| is V11() real ext-real non negative Element of REAL
p is non empty ordinal natural V11() real ext-real positive non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of NAT
TOP-REAL p is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable L15()
the carrier of (TOP-REAL p) is non empty functional set
(p) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL p
0. (TOP-REAL p) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V54( TOP-REAL p) V124() V125() V126() Element of the carrier of (TOP-REAL p)
the ZeroF of (TOP-REAL p) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
(p,(0. (TOP-REAL p)),1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL p
Ball ((0. (TOP-REAL p)),1) is non empty functional open V162( TOP-REAL p) Element of bool the carrier of (TOP-REAL p)
bool the carrier of (TOP-REAL p) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p) : not 1 <= |.(b₁ - (0. (TOP-REAL p))).| } is set
(TOP-REAL p) | (Ball ((0. (TOP-REAL p)),1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL p
p1 is non empty V11() real ext-real positive non negative set
U1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
(p,U1,p1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL p
Ball (U1,p1) is non empty functional open V162( TOP-REAL p) Element of bool the carrier of (TOP-REAL p)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p) : not p1 <= |.(b₁ - U1).| } is set
(TOP-REAL p) | (Ball (U1,p1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL p
(T) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
0. (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V54( TOP-REAL T) V124() V125() V126() Element of the carrier of (TOP-REAL T)
the ZeroF of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(T,(0. (TOP-REAL T)),1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball ((0. (TOP-REAL T)),1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not 1 <= |.(b₁ - (0. (TOP-REAL T))).| } is set
(TOP-REAL T) | (Ball ((0. (TOP-REAL T)),1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the carrier of (T) is non empty set
the carrier of (p,U1,p1) is non empty set
S2 is Element of the carrier of (T)
X2 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
p1 * X2 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
(p1 * X2) + U1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
the U7 of (TOP-REAL p) is non empty Relation-like [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] -defined the carrier of (TOP-REAL p) -valued Function-like V26([: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):]
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):] is non empty set
K155( the carrier of (TOP-REAL p), the U7 of (TOP-REAL p),(p1 * X2),U1) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
((p1 * X2) + U1) - U1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
- U1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
K184((TOP-REAL p),((p1 * X2) + U1),(- U1)) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
K155( the carrier of (TOP-REAL p), the U7 of (TOP-REAL p),((p1 * X2) + U1),(- U1)) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
|.(((p1 * X2) + U1) - U1).| is V11() real ext-real non negative Element of REAL
|.(p1 * X2).| is V11() real ext-real non negative Element of REAL
abs p1 is non empty V11() real ext-real positive non negative Element of REAL
|.X2.| is V11() real ext-real non negative Element of REAL
(abs p1) * |.X2.| is V11() real ext-real non negative Element of REAL
p1 * |.X2.| is V11() real ext-real non negative Element of REAL
p1 * 1 is non empty V11() real ext-real positive non negative Element of REAL
f is Element of the carrier of (p,U1,p1)
[: the carrier of (T), the carrier of (p,U1,p1):] is non empty Relation-like set
bool [: the carrier of (T), the carrier of (p,U1,p1):] is non empty set
S2 is non empty Relation-like the carrier of (T) -defined the carrier of (p,U1,p1) -valued Function-like V26( the carrier of (T)) quasi_total Element of bool [: the carrier of (T), the carrier of (p,U1,p1):]
X2 is Element of the carrier of (T)
S2 . X2 is Element of the carrier of (p,U1,p1)
U is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
p1 * U is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
(p1 * U) + U1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
the U7 of (TOP-REAL p) is non empty Relation-like [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] -defined the carrier of (TOP-REAL p) -valued Function-like V26([: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):]
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):] is non empty set
K155( the carrier of (TOP-REAL p), the U7 of (TOP-REAL p),(p1 * U),U1) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
f is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
p1 * f is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
(p1 * f) + U1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
K155( the carrier of (TOP-REAL p), the U7 of (TOP-REAL p),(p1 * f),U1) is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
X2 is non empty Relation-like the carrier of (T) -defined the carrier of (p,U1,p1) -valued Function-like V26( the carrier of (T)) quasi_total Element of bool [: the carrier of (T), the carrier of (p,U1,p1):]
p1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
(p,p1,X) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL p
Ball (p1,X) is non empty functional open V162( TOP-REAL p) Element of bool the carrier of (TOP-REAL p)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p) : not X <= |.(b₁ - p1).| } is set
(TOP-REAL p) | (Ball (p1,X)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL p
U1 is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p)
(p,U1,X1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL p
Ball (U1,X1) is non empty functional open V162( TOP-REAL p) Element of bool the carrier of (TOP-REAL p)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() p -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL p) : not X1 <= |.(b₁ - U1).| } is set
(TOP-REAL p) | (Ball (U1,X1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL p
(T) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
0. (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V54( TOP-REAL T) V124() V125() V126() Element of the carrier of (TOP-REAL T)
the ZeroF of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(T,(0. (TOP-REAL T)),1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball ((0. (TOP-REAL T)),1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not 1 <= |.(b₁ - (0. (TOP-REAL T))).| } is set
(TOP-REAL T) | (Ball ((0. (TOP-REAL T)),1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
bool the carrier of (TOP-REAL T) is non empty set
[#] (TOP-REAL T) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL T)
c₂ is non empty functional open (T) Element of bool the carrier of (TOP-REAL T)
M is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
X is V11() real ext-real set
Ball (M,X) is functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not X <= |.(b₁ - M).| } is set
(T,M,X) is TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
(TOP-REAL T) | (Ball (M,X)) is strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
(TOP-REAL T) | ([#] (TOP-REAL T)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
(T) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
0. (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V54( TOP-REAL T) V124() V125() V126() Element of the carrier of (TOP-REAL T)
the ZeroF of (TOP-REAL T) is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
(T,(0. (TOP-REAL T)),1) is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
Ball ((0. (TOP-REAL T)),1) is non empty functional open Element of bool the carrier of (TOP-REAL T)
{ b₁ where b₁ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T) : not 1 <= |.(b₁ - (0. (TOP-REAL T))).| } is set
(TOP-REAL T) | (Ball ((0. (TOP-REAL T)),1)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
X1 is non empty TopSpace-like TopStruct
T is non empty TopSpace-like TopStruct
the carrier of T is non empty set
c₂ is non empty TopSpace-like TopStruct
the carrier of c₂ is non empty set
bool the carrier of c₂ is non empty set
bool the carrier of T is non empty set
M is Element of the carrier of T
X is a_neighborhood of M
X1 is open Element of bool the carrier of c₂
T | X is strict TopSpace-like SubSpace of T
c₂ | X1 is strict TopSpace-like SubSpace of c₂
the carrier of (T | X) is set
the carrier of (c₂ | X1) is set
[: the carrier of (T | X), the carrier of (c₂ | X1):] is Relation-like set
bool [: the carrier of (T | X), the carrier of (c₂ | X1):] is non empty set
p is Relation-like the carrier of (T | X) -defined the carrier of (c₂ | X1) -valued Function-like quasi_total Element of bool [: the carrier of (T | X), the carrier of (c₂ | X1):]
p1 is Element of bool the carrier of T
[#] (T | X) is non proper open closed dense Element of bool the carrier of (T | X)
bool the carrier of (T | X) is non empty set
S1 is non empty TopStruct
the carrier of S1 is non empty set
U2 is non empty TopStruct
the carrier of U2 is non empty set
[: the carrier of S1, the carrier of U2:] is non empty Relation-like set
bool [: the carrier of S1, the carrier of U2:] is non empty set
rng p is Element of bool the carrier of (c₂ | X1)
bool the carrier of (c₂ | X1) is non empty set
[#] (c₂ | X1) is non proper open closed dense Element of bool the carrier of (c₂ | X1)
U1 is Element of bool the carrier of (T | X)
p .: U1 is Element of bool the carrier of (c₂ | X1)
(T | X) | U1 is strict TopSpace-like SubSpace of T | X
(c₂ | X1) | (p .: U1) is strict TopSpace-like SubSpace of c₂ | X1
U is TopSpace-like open SubSpace of c₂
the carrier of U is set
bool the carrier of U is non empty set
bool the carrier of S1 is non empty set
S2 is non empty Relation-like the carrier of S1 -defined the carrier of U2 -valued Function-like V26( the carrier of S1) quasi_total Element of bool [: the carrier of S1, the carrier of U2:]
U3 is Element of bool the carrier of S1
S2 .: U3 is Element of bool the carrier of U2
bool the carrier of U2 is non empty set
M2 is Element of bool the carrier of S1
S2 .: M2 is Element of bool the carrier of U2
X2 is open Element of bool the carrier of T
the topology of T is non empty open Element of bool (bool the carrier of T)
bool (bool the carrier of T) is non empty set
[#] S1 is non empty non proper dense Element of bool the carrier of S1
X2 /\ ([#] S1) is Element of bool the carrier of S1
the topology of S1 is open Element of bool (bool the carrier of S1)
bool (bool the carrier of S1) is non empty set
U is Element of bool the carrier of U
f is Element of bool the carrier of c₂
U3 is open Element of bool the carrier of c₂
T | X2 is strict TopSpace-like SubSpace of T
p .: X is Element of bool the carrier of (c₂ | X1)
p .: X2 is Element of bool the carrier of (c₂ | X1)
c₂ | U3 is strict TopSpace-like SubSpace of c₂
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() L15()
the carrier of (TOP-REAL T) is non empty functional set
c₂ is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
bool the carrier of (TOP-REAL T) is non empty set
[#] (TOP-REAL T) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL T)
Int ([#] (TOP-REAL T)) is functional open Element of bool the carrier of (TOP-REAL T)
M is functional a_neighborhood of c₂
X is functional open Element of bool the carrier of (TOP-REAL T)
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() (T) L15()
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() (T) L15()
the carrier of (TOP-REAL T) is non empty functional set
bool the carrier of (TOP-REAL T) is non empty set
c₂ is non empty TopSpace-like TopStruct
the carrier of c₂ is non empty set
M is Element of the carrier of c₂
X is a_neighborhood of M
X1 is functional open Element of bool the carrier of (TOP-REAL T)
bool the carrier of c₂ is non empty set
p is open Element of bool the carrier of c₂
p1 is functional open Element of bool the carrier of (TOP-REAL T)
U1 is non empty Element of bool the carrier of c₂
c₂ | U1 is non empty strict TopSpace-like SubSpace of c₂
(TOP-REAL T) | p1 is strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the carrier of (c₂ | U1) is non empty set
the carrier of ((TOP-REAL T) | p1) is set
[: the carrier of (c₂ | U1), the carrier of ((TOP-REAL T) | p1):] is Relation-like set
bool [: the carrier of (c₂ | U1), the carrier of ((TOP-REAL T) | p1):] is non empty set
S1 is Relation-like the carrier of (c₂ | U1) -defined the carrier of ((TOP-REAL T) | p1) -valued Function-like quasi_total Element of bool [: the carrier of (c₂ | U1), the carrier of ((TOP-REAL T) | p1):]
[#] (c₂ | U1) is non empty non proper open closed dense non boundary Element of bool the carrier of (c₂ | U1)
bool the carrier of (c₂ | U1) is non empty set
S2 is non empty TopSpace-like SubSpace of c₂
the carrier of S2 is non empty set
U2 is non empty TopSpace-like TopStruct
the carrier of U2 is non empty set
[: the carrier of S2, the carrier of U2:] is non empty Relation-like set
bool [: the carrier of S2, the carrier of U2:] is non empty set
[#] ((TOP-REAL T) | p1) is non proper open closed dense Element of bool the carrier of ((TOP-REAL T) | p1)
bool the carrier of ((TOP-REAL T) | p1) is non empty set
[#] (TOP-REAL T) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL T)
X2 is non empty Relation-like the carrier of S2 -defined the carrier of U2 -valued Function-like V26( the carrier of S2) quasi_total Element of bool [: the carrier of S2, the carrier of U2:]
X2 . M is set
U is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
f is functional open (T) Element of bool the carrier of (TOP-REAL T)
X2 " is non empty Relation-like the carrier of U2 -defined the carrier of S2 -valued Function-like V26( the carrier of U2) quasi_total Element of bool [: the carrier of U2, the carrier of S2:]
[: the carrier of U2, the carrier of S2:] is non empty Relation-like set
bool [: the carrier of U2, the carrier of S2:] is non empty set
bool the carrier of U2 is non empty set
U is non empty functional open (T) Element of bool the carrier of (TOP-REAL T)
rng X2 is non empty Element of bool the carrier of U2
[#] U2 is non empty non proper open closed dense non boundary Element of bool the carrier of U2
dom (X2 ") is non empty Element of bool the carrier of U2
(X2 ") . U is set
U3 is non empty Element of bool the carrier of U2
(X2 ") .: U3 is Element of bool the carrier of S2
bool the carrier of S2 is non empty set
(X2 ") | U3 is Relation-like the carrier of U2 -defined U3 -defined the carrier of U2 -defined the carrier of S2 -valued Function-like Element of bool [: the carrier of U2, the carrier of S2:]
dom ((X2 ") | U3) is Element of bool U3
bool U3 is non empty set
U2 | U3 is non empty strict TopSpace-like SubSpace of U2
[#] (U2 | U3) is non empty non proper open closed dense non boundary Element of bool the carrier of (U2 | U3)
the carrier of (U2 | U3) is non empty set
bool the carrier of (U2 | U3) is non empty set
rng ((X2 ") | U3) is Element of bool the carrier of S2
M2 is non empty Element of bool the carrier of S2
S2 | M2 is non empty strict TopSpace-like SubSpace of S2
[#] (S2 | M2) is non empty non proper open closed dense non boundary Element of bool the carrier of (S2 | M2)
the carrier of (S2 | M2) is non empty set
bool the carrier of (S2 | M2) is non empty set
[: the carrier of (U2 | U3), the carrier of (S2 | M2):] is non empty Relation-like set
bool [: the carrier of (U2 | U3), the carrier of (S2 | M2):] is non empty set
T2 is non empty Relation-like the carrier of (U2 | U3) -defined the carrier of (S2 | M2) -valued Function-like V26( the carrier of (U2 | U3)) quasi_total Element of bool [: the carrier of (U2 | U3), the carrier of (S2 | M2):]
S is Relation-like Function-like set
dom S is set
S " is Relation-like Function-like set
f2 is Element of the carrier of S2
[#] S2 is non empty non proper open closed dense non boundary Element of bool the carrier of S2
[#] c₂ is non empty non proper open closed dense non boundary Element of bool the carrier of c₂
the topology of (TOP-REAL T) is non empty open Element of bool (bool the carrier of (TOP-REAL T))
bool (bool the carrier of (TOP-REAL T)) is non empty set
U /\ ([#] U2) is Element of bool the carrier of U2
the topology of U2 is non empty open Element of bool (bool the carrier of U2)
bool (bool the carrier of U2) is non empty set
Q is Element of bool the carrier of c₂
S is a_neighborhood of M
rng (X2 ") is non empty Element of bool the carrier of S2
c₂ | S is strict TopSpace-like SubSpace of c₂
(TOP-REAL T) | U is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() (T) L15()
the carrier of (TOP-REAL T) is non empty functional set
bool the carrier of (TOP-REAL T) is non empty set
c₂ is non empty TopSpace-like TopStruct
the carrier of c₂ is non empty set
M is Element of the carrier of c₂
X is a_neighborhood of M
X1 is non empty functional open (T) Element of bool the carrier of (TOP-REAL T)
p is functional open (T) Element of bool the carrier of (TOP-REAL T)
p1 is a_neighborhood of M
U1 is functional open (T) Element of bool the carrier of (TOP-REAL T)
M is Element of the carrier of c₂
X is a_neighborhood of M
X1 is functional open (T) Element of bool the carrier of (TOP-REAL T)
p is functional open Element of bool the carrier of (TOP-REAL T)
p1 is a_neighborhood of M
U1 is functional open Element of bool the carrier of (TOP-REAL T)
T is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() (T) L15()
[#] (TOP-REAL T) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL T)
the carrier of (TOP-REAL T) is non empty functional set
bool the carrier of (TOP-REAL T) is non empty set
c₂ is non empty TopSpace-like TopStruct
the carrier of c₂ is non empty set
M is Element of the carrier of c₂
X is a_neighborhood of M
X1 is non empty functional open (T) Element of bool the carrier of (TOP-REAL T)
(TOP-REAL T) | ([#] (TOP-REAL T)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the topology of (TOP-REAL T) is non empty open Element of bool (bool the carrier of (TOP-REAL T))
bool (bool the carrier of (TOP-REAL T)) is non empty set
TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #) is non empty strict TopSpace-like V204() second-countable TopStruct
(TOP-REAL T) | X1 is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
p is a_neighborhood of M
c₂ | p is strict TopSpace-like SubSpace of c₂
p1 is non empty TopSpace-like TopStruct
U1 is non empty TopSpace-like TopStruct
M is Element of the carrier of c₂
X is a_neighborhood of M
the non empty functional open (T) Element of bool the carrier of (TOP-REAL T) is non empty functional open (T) Element of bool the carrier of (TOP-REAL T)
p is functional open Element of bool the carrier of (TOP-REAL T)
p1 is a_neighborhood of M
c₂ | p1 is strict TopSpace-like SubSpace of c₂
U1 is functional open Element of bool the carrier of (TOP-REAL T)
(TOP-REAL T) | U1 is strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
U2 is non empty TopSpace-like TopStruct
S1 is non empty TopSpace-like TopStruct
T is ordinal natural V11() real ext-real non negative V35() cardinal set
c₂ is non empty TopSpace-like TopStruct
the carrier of c₂ is non empty set
bool the carrier of c₂ is non empty set
bool (bool the carrier of c₂) is non empty set
M is Element of the carrier of c₂
TOP-REAL T is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() (T) L15()
[#] (TOP-REAL T) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL T)
the carrier of (TOP-REAL T) is non empty functional set
bool the carrier of (TOP-REAL T) is non empty set
X is a_neighborhood of M
c₂ | X is strict TopSpace-like SubSpace of c₂
(TOP-REAL T) | ([#] (TOP-REAL T)) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL T
the topology of (TOP-REAL T) is non empty open Element of bool (bool the carrier of (TOP-REAL T))
bool (bool the carrier of (TOP-REAL T)) is non empty set
TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #) is non empty strict TopSpace-like V204() second-countable TopStruct
the carrier of (c₂ | X) is set
the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #) is non empty set
[: the carrier of (c₂ | X), the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #):] is Relation-like set
bool [: the carrier of (c₂ | X), the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #):] is non empty set
X1 is Relation-like the carrier of (c₂ | X) -defined the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #) -valued Function-like V26( the carrier of (c₂ | X)) quasi_total Element of bool [: the carrier of (c₂ | X), the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #):]
dom X1 is Element of bool the carrier of (c₂ | X)
bool the carrier of (c₂ | X) is non empty set
[#] (c₂ | X) is non proper open closed dense Element of bool the carrier of (c₂ | X)
rng X1 is Element of bool the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #)
bool the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #) is non empty set
[#] TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #) is non empty non proper open closed dense non boundary Element of bool the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #)
X1 " is Relation-like the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #) -defined the carrier of (c₂ | X) -valued Function-like quasi_total Element of bool [: the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #), the carrier of (c₂ | X):]
[: the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #), the carrier of (c₂ | X):] is Relation-like set
bool [: the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #), the carrier of (c₂ | X):] is non empty set
Int X is open Element of bool the carrier of c₂
X1 . M is set
p1 is ordinal natural V11() real ext-real non negative V33() V34() V35() cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of NAT
TOP-REAL p1 is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable (p1) L15()
p is Relation-like NAT -defined Function-like V35() T -element V43() V44() V124() V125() V126() Element of the carrier of (TOP-REAL T)
U1 is open V301( TOP-REAL T,p) Element of bool (bool the carrier of (TOP-REAL T))
S1 is non empty set
{ H₁(b₁) where b₁ is Element of S1 : b₁ in U1 } is set
card U1 is ordinal cardinal set
card { H₁(b₁) where b₁ is Element of S1 : b₁ in U1 } is ordinal cardinal set
S2 is set
X2 is Element of S1
(X1 ") .: X2 is Element of bool the carrier of (c₂ | X)
((X1 ") .: X2) /\ (Int X) is Element of bool the carrier of c₂
S2 is Element of bool (bool the carrier of c₂)
X2 is Element of bool the carrier of c₂
U is Element of S1
(X1 ") .: U is Element of bool the carrier of (c₂ | X)
((X1 ") .: U) /\ (Int X) is Element of bool the carrier of c₂
f is Element of bool the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #)
U is functional Element of bool the carrier of (TOP-REAL T)
U3 is non empty TopStruct
the carrier of U3 is non empty set
[: the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #), the carrier of U3:] is non empty Relation-like set
bool [: the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #), the carrier of U3:] is non empty set
M2 is non empty Relation-like the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #) -defined the carrier of U3 -valued Function-like V26( the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #)) quasi_total Element of bool [: the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #), the carrier of U3:]
M2 .: f is Element of bool the carrier of U3
bool the carrier of U3 is non empty set
the topology of (c₂ | X) is non empty open Element of bool (bool the carrier of (c₂ | X))
bool (bool the carrier of (c₂ | X)) is non empty set
the topology of c₂ is non empty open Element of bool (bool the carrier of c₂)
T2 is Element of bool the carrier of c₂
T2 /\ ([#] (c₂ | X)) is Element of bool the carrier of (c₂ | X)
X /\ (Int X) is Element of bool the carrier of c₂
T2 /\ (X /\ (Int X)) is Element of bool the carrier of c₂
T2 /\ (Int X) is Element of bool the carrier of c₂
X2 is set
U is Element of S1
(X1 ") .: U is Element of bool the carrier of (c₂ | X)
((X1 ") .: U) /\ (Int X) is Element of bool the carrier of c₂
f is Relation-like Function-like set
f . M is set
[M,(f . M)] is set
{M,(f . M)} is non empty set
{M} is non empty trivial 1 -element set
{{M,(f . M)},{M}} is non empty set
[p,M] is set
{p,M} is non empty set
{p} is non empty trivial functional 1 -element set
{{p,M},{p}} is non empty set
f ~ is Relation-like set
f " is Relation-like Function-like set
Intersect S2 is Element of bool the carrier of c₂
X2 is Element of bool the carrier of c₂
X2 /\ ([#] (c₂ | X)) is Element of bool the carrier of (c₂ | X)
the topology of c₂ is non empty open Element of bool (bool the carrier of c₂)
the topology of (c₂ | X) is non empty open Element of bool (bool the carrier of (c₂ | X))
bool (bool the carrier of (c₂ | X)) is non empty set
f is non empty TopStruct
the carrier of f is non empty set
bool the carrier of f is non empty set
U is Element of bool the carrier of f
X1 .: U is Element of bool the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #)
the topology of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #) is non empty open Element of bool (bool the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #))
bool (bool the carrier of TopStruct(# the carrier of (TOP-REAL T), the topology of (TOP-REAL T) #)) is non empty set
U3 is functional Element of bool the carrier of (TOP-REAL T)
[M,p] is set
{M,p} is non empty set
{M} is non empty trivial 1 -element set
{{M,p},{M}} is non empty set
T2 is functional Element of bool the carrier of (TOP-REAL T)
f2 is Element of S1
(X1 ") .: f2 is Element of bool the carrier of (c₂ | X)
((X1 ") .: f2) /\ (Int X) is Element of bool the carrier of c₂
Q is Element of bool the carrier of c₂
M2 is Relation-like Function-like set
M2 " is Relation-like Function-like set
(X1 ") .: (X1 .: U) is Element of bool the carrier of (c₂ | X)
X2 is open V301(c₂,M) Element of bool (bool the carrier of c₂)
TOP-REAL {} is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable ( {} ) L15()
[#] (TOP-REAL {}) is non empty non proper functional open closed dense non boundary Element of bool the carrier of (TOP-REAL {})
the carrier of (TOP-REAL {}) is non empty functional set
bool the carrier of (TOP-REAL {}) is non empty set
(TOP-REAL {}) | ([#] (TOP-REAL {})) is non empty strict TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL {}
the topology of (TOP-REAL {}) is non empty open Element of bool (bool the carrier of (TOP-REAL {}))
bool (bool the carrier of (TOP-REAL {})) is non empty set
TopStruct(# the carrier of (TOP-REAL {}), the topology of (TOP-REAL {}) #) is non empty strict TopSpace-like V204() second-countable TopStruct
c₂ is non empty TopSpace-like TopStruct
the carrier of c₂ is non empty set
bool the carrier of c₂ is non empty set
M is Element of bool the carrier of c₂
X is Element of the carrier of c₂
{X} is non empty trivial 1 -element set
X1 is a_neighborhood of X
Int X1 is open Element of bool the carrier of c₂
c₂ | X1 is strict TopSpace-like SubSpace of c₂
the carrier of (c₂ | X1) is set
the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))) is non empty set
[: the carrier of (c₂ | X1), the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))):] is Relation-like set
bool [: the carrier of (c₂ | X1), the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))):] is non empty set
p is Relation-like the carrier of (c₂ | X1) -defined the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))) -valued Function-like V26( the carrier of (c₂ | X1)) quasi_total Element of bool [: the carrier of (c₂ | X1), the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))):]
p1 is Element of bool the carrier of c₂
U1 is set
[#] (c₂ | X1) is non proper open closed dense Element of bool the carrier of (c₂ | X1)
bool the carrier of (c₂ | X1) is non empty set
dom p is Element of bool the carrier of (c₂ | X1)
p . U1 is set
rng p is Element of bool the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {})))
bool the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))) is non empty set
p . X is set
0. (TOP-REAL {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V54( TOP-REAL {}) V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
the ZeroF of (TOP-REAL {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
[#] ((TOP-REAL {}) | ([#] (TOP-REAL {}))) is non empty non proper open closed dense non boundary Element of bool the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {})))
the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))) is non empty set
bool the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))) is non empty set
0. (TOP-REAL {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V54( TOP-REAL {}) V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
the ZeroF of (TOP-REAL {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V35() cardinal {} -element V43() V44() V45() V124() V125() V126() V127() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V140() bounded_below bounded_above real-bounded V317() Element of the carrier of (TOP-REAL {})
{(0. (TOP-REAL {}))} is non empty trivial functional 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
c₂ is non empty TopSpace-like TopStruct
the carrier of c₂ is non empty set
M is Element of the carrier of c₂
bool the carrier of c₂ is non empty set
{M} is non empty trivial 1 -element set
X is Element of bool the carrier of c₂
X1 is a_neighborhood of M
[M,(0. (TOP-REAL {}))] is set
{M,(0. (TOP-REAL {}))} is non empty set
{{M,(0. (TOP-REAL {}))},{M}} is non empty set
{[M,(0. (TOP-REAL {}))]} is non empty trivial Relation-like Function-like one-to-one constant 1 -element set
c₂ | X1 is strict TopSpace-like SubSpace of c₂
[#] (c₂ | X1) is non proper open closed dense Element of bool the carrier of (c₂ | X1)
the carrier of (c₂ | X1) is set
bool the carrier of (c₂ | X1) is non empty set
[: the carrier of (c₂ | X1), the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))):] is Relation-like set
dom {[M,(0. (TOP-REAL {}))]} is non empty trivial 1 -element set
bool [: the carrier of (c₂ | X1), the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))):] is non empty set
p1 is Relation-like the carrier of (c₂ | X1) -defined the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))) -valued Function-like V26( the carrier of (c₂ | X1)) quasi_total Element of bool [: the carrier of (c₂ | X1), the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {}))):]
rng p1 is Element of bool the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {})))
U1 is Element of bool the carrier of (c₂ | X1)
p1 .: U1 is Element of bool the carrier of ((TOP-REAL {}) | ([#] (TOP-REAL {})))
c₂ is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL c₂ is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() first-countable (c₂) L15()
the carrier of (TOP-REAL c₂) is non empty functional set
the topology of (TOP-REAL c₂) is non empty open Element of bool (bool the carrier of (TOP-REAL c₂))
bool the carrier of (TOP-REAL c₂) is non empty set
bool (bool the carrier of (TOP-REAL c₂)) is non empty set
TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #) is non empty strict TopSpace-like V204() second-countable TopStruct
the carrier of TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #) is non empty set
bool the carrier of TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #) is non empty set
bool (bool the carrier of TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #)) is non empty set
{ (card b₁) where b₁ is open V294( TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #)) Element of bool (bool the carrier of TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #)) : verum } is set
{ (card b₁) where b₁ is open V294( TOP-REAL c₂) Element of bool (bool the carrier of (TOP-REAL c₂)) : verum } is set
X is set
X1 is open V294( TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #)) Element of bool (bool the carrier of TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #))
card X1 is ordinal cardinal set
p is open V294( TOP-REAL c₂) Element of bool (bool the carrier of (TOP-REAL c₂))
card p is ordinal cardinal set
X1 is open V294( TOP-REAL c₂) Element of bool (bool the carrier of (TOP-REAL c₂))
card X1 is ordinal cardinal set
p is open V294( TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #)) Element of bool (bool the carrier of TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #))
card p is ordinal cardinal set
weight TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #) is ordinal cardinal set
meet { (card b₁) where b₁ is open V294( TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #)) Element of bool (bool the carrier of TopStruct(# the carrier of (TOP-REAL c₂), the topology of (TOP-REAL c₂) #)) : verum } is set
weight (TOP-REAL c₂) is ordinal cardinal set
meet { (card b₁) where b₁ is open V294( TOP-REAL c₂) Element of bool (bool the carrier of (TOP-REAL c₂)) : verum } is set
c₂ is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL c₂ is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() V204() second-countable first-countable (c₂) L15()
c₂ is ordinal natural V11() real ext-real non negative V35() cardinal set
TOP-REAL c₂ is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() V204() second-countable first-countable (c₂) L15()
c₂ is ordinal natural V11() real ext-real non negative V35() cardinal set
M is non empty TopSpace-like TopStruct
M is non empty TopSpace-like TopStruct
M is non empty TopSpace-like TopStruct
c₂ is non empty TopSpace-like TopStruct
c₂ is ordinal natural V11() real ext-real non negative V35() cardinal set
M is non empty TopSpace-like T_0 T_1 T_2 V204() second-countable first-countable (c₂) (c₂) () TopStruct
X is non empty TopSpace-like T_0 T_1 T_2 SubSpace of M
[#] X is non empty non proper open closed dense non boundary Element of bool the carrier of X
the carrier of X is non empty set
bool the carrier of X is non empty set
[#] M is non empty non proper open closed dense non boundary Element of bool the carrier of M
the carrier of M is non empty set
bool the carrier of M is non empty set
X1 is non empty open Element of bool the carrier of M
M | X1 is non empty strict TopSpace-like T_0 T_1 T_2 V204() second-countable SubSpace of M
[#] (M | X1) is non empty non proper open closed dense non boundary Element of bool the carrier of (M | X1)
the carrier of (M | X1) is non empty set
bool the carrier of (M | X1) is non empty set
the topology of X is non empty open Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty set
TopStruct(# the carrier of X, the topology of X #) is non empty strict TopSpace-like TopStruct
the topology of (M | X1) is non empty open Element of bool (bool the carrier of (M | X1))
bool (bool the carrier of (M | X1)) is non empty set
TopStruct(# the carrier of (M | X1), the topology of (M | X1) #) is non empty strict TopSpace-like TopStruct
{ (card b₁) where b₁ is open V294(X) Element of bool (bool the carrier of X) : verum } is set
{ (card b₁) where b₁ is open V294(M | X1) Element of bool (bool the carrier of (M | X1)) : verum } is set
p is set
p1 is open V294(X) Element of bool (bool the carrier of X)
card p1 is ordinal cardinal set
U1 is open V294(M | X1) Element of bool (bool the carrier of (M | X1))
card U1 is ordinal cardinal set
p1 is open V294(M | X1) Element of bool (bool the carrier of (M | X1))
card p1 is ordinal cardinal set
U1 is open V294(X) Element of bool (bool the carrier of X)
card U1 is ordinal cardinal set
weight X is ordinal cardinal set
meet { (card b₁) where b₁ is open V294(X) Element of bool (bool the carrier of X) : verum } is set
weight (M | X1) is ordinal cardinal set
meet { (card b₁) where b₁ is open V294(M | X1) Element of bool (bool the carrier of (M | X1)) : verum } is set
TOP-REAL c₂ is non empty TopSpace-like T_0 T_1 T_2 V100() V146() V147() V148() V149() V150() V151() V152() V158() V204() second-countable first-countable (c₂) (c₂) () L15()
the carrier of (TOP-REAL c₂) is non empty functional set
bool the carrier of (TOP-REAL c₂) is non empty set
p is Element of the carrier of X
p1 is Element of the carrier of M
U1 is a_neighborhood of p1
S1 is functional open Element of bool the carrier of (TOP-REAL c₂)
U2 is open Element of bool the carrier of M
S2 is functional open Element of bool the carrier of (TOP-REAL c₂)
X2 is non empty TopSpace-like T_0 T_1 T_2 open SubSpace of M
the carrier of X2 is non empty set
bool the carrier of X2 is non empty set
U2 /\ X1 is open Element of bool the carrier of M
M | U2 is strict TopSpace-like T_0 T_1 T_2 V204() second-countable SubSpace of M
(TOP-REAL c₂) | S2 is strict TopSpace-like T_0 T_1 T_2 V204() second-countable SubSpace of TOP-REAL c₂
the carrier of (M | U2) is set
the carrier of ((TOP-REAL c₂) | S2) is set
[: the carrier of (M | U2), the carrier of ((TOP-REAL c₂) | S2):] is Relation-like set
bool [: the carrier of (M | U2), the carrier of ((TOP-REAL c₂) | S2):] is non empty set
f is Relation-like the carrier of (M | U2) -defined the carrier of ((TOP-REAL c₂) | S2) -valued Function-like quasi_total Element of bool [: the carrier of (M | U2), the carrier of ((TOP-REAL c₂) | S2):]
U is Element of bool the carrier of X2
U is a_neighborhood of p
[#] (M | U2) is non proper open closed dense Element of bool the carrier of (M | U2)
bool the carrier of (M | U2) is non empty set
U3 is Element of bool the carrier of (M | U2)
f .: U3 is Element of bool the carrier of ((TOP-REAL c₂) | S2)
bool the carrier of ((TOP-REAL c₂) | S2) is non empty set
(M | U2) | U3 is strict TopSpace-like T_0 T_1 T_2 V204() second-countable SubSpace of M | U2
((TOP-REAL c₂) | S2) | (f .: U3) is strict TopSpace-like T_0 T_1 T_2 V204() second-countable SubSpace of (TOP-REAL c₂) | S2
M2 is non empty TopSpace-like T_0 T_1 T_2 SubSpace of M
the carrier of M2 is non empty set
T2 is non empty TopSpace-like T_0 T_1 T_2 SubSpace of TOP-REAL c₂
the carrier of T2 is non empty set
[: the carrier of M2, the carrier of T2:] is non empty Relation-like set
bool [: the carrier of M2, the carrier of T2:] is non empty set
bool the carrier of M2 is non empty set
f2 is non empty Relation-like the carrier of M2 -defined the carrier of T2 -valued Function-like V26( the carrier of M2) quasi_total Element of bool [: the carrier of M2, the carrier of T2:]
S is Element of bool the carrier of M2
f2 .: S is Element of bool the carrier of T2
bool the carrier of T2 is non empty set
Q is Element of bool the carrier of M2
f2 .: Q is Element of bool the carrier of T2
[#] ((TOP-REAL c₂) | S2) is non proper open closed dense Element of bool the carrier of ((TOP-REAL c₂) | S2)
the topology of M is non empty open Element of bool (bool the carrier of M)
bool (bool the carrier of M) is non empty set
[#] M2 is non empty non proper open closed dense non boundary Element of bool the carrier of M2
the topology of M2 is non empty open Element of bool (bool the carrier of M2)
bool (bool the carrier of M2) is non empty set
bool the carrier of T2 is non empty set
S is open Element of bool the carrier of T2
the topology of T2 is non empty open Element of bool (bool the carrier of T2)
bool (bool the carrier of T2) is non empty set
the topology of (TOP-REAL c₂) is non empty open Element of bool (bool the carrier of (TOP-REAL c₂))
bool (bool the carrier of (TOP-REAL c₂)) is non empty set
[#] T2 is non empty non proper open closed dense non boundary Element of bool the carrier of T2
Q is functional Element of bool the carrier of (TOP-REAL c₂)
Q /\ ([#] T2) is Element of bool the carrier of T2
S is functional open Element of bool the carrier of (TOP-REAL c₂)
U4 is Element of bool the carrier of (M | X1)
(M | X1) | U4 is strict TopSpace-like T_0 T_1 T_2 V204() second-countable SubSpace of M | X1
U5 is Element of bool the carrier of M
M | U5 is strict TopSpace-like T_0 T_1 T_2 V204() second-countable SubSpace of M
X | U is strict TopSpace-like T_0 T_1 T_2 SubSpace of X
the carrier of (X | U) is set
the topology of (X | U) is non empty open Element of bool (bool the carrier of (X | U))
bool the carrier of (X | U) is non empty set
bool (bool the carrier of (X | U)) is non empty set
TopStruct(# the carrier of (X | U), the topology of (X | U) #) is strict TopSpace-like TopStruct
the carrier of ((M | U2) | U3) is set
the topology of ((M | U2) | U3) is non empty open Element of bool (bool the carrier of ((M | U2) | U3))
bool the carrier of ((M | U2) | U3) is non empty set
bool (bool the carrier of ((M | U2) | U3)) is non empty set
TopStruct(# the carrier of ((M | U2) | U3), the topology of ((M | U2) | U3) #) is strict TopSpace-like TopStruct
(TOP-REAL c₂) | S is strict TopSpace-like T_0 T_1 T_2 V204() second-countable SubSpace of TOP-REAL c₂