:: BORSUK_7 semantic presentation

REAL is non empty V32() V195() V196() V197() V201() set
NAT is V195() V196() V197() V198() V199() V200() V201() Element of bool REAL
bool REAL is V74() set
I[01] is non empty strict TopSpace-like V224() V231() pathwise_connected connected SubSpace of R^1
R^1 is non empty strict TopSpace-like V231() pathwise_connected connected TopStruct
the carrier of I[01] is non empty V195() V196() V197() set
[:I[01],I[01]:] is non empty strict TopSpace-like TopStruct
the carrier of [:I[01],I[01]:] is non empty set
COMPLEX is non empty V32() V195() V201() set
RAT is non empty V32() V195() V196() V197() V198() V201() set
INT is non empty V32() V195() V196() V197() V198() V199() V201() set
[:COMPLEX,COMPLEX:] is Relation-like complex-yielding set
bool [:COMPLEX,COMPLEX:] is set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like complex-yielding set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is set
[:REAL,REAL:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:REAL,REAL:] is set
[:[:REAL,REAL:],REAL:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:[:REAL,REAL:],REAL:] is set
[:RAT,RAT:] is Relation-like RAT -valued complex-yielding ext-real-valued real-valued set
bool [:RAT,RAT:] is set
[:[:RAT,RAT:],RAT:] is Relation-like RAT -valued complex-yielding ext-real-valued real-valued set
bool [:[:RAT,RAT:],RAT:] is set
[:INT,INT:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
bool [:INT,INT:] is set
[:[:INT,INT:],INT:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
bool [:[:INT,INT:],INT:] is set
[:NAT,NAT:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
bool [:[:NAT,NAT:],NAT:] is set
NAT is V195() V196() V197() V198() V199() V200() V201() set
bool NAT is set
bool NAT is set
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V32() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued set
the Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V32() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued set is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V32() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued set
1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
{{},1} is non empty set
[:NAT,REAL:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:NAT,REAL:] is set
[:NAT,COMPLEX:] is Relation-like complex-yielding set
bool [:NAT,COMPLEX:] is set
[:1,1:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
bool [:1,1:] is set
[:[:1,1:],1:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
bool [:[:1,1:],1:] is set
[:[:1,1:],REAL:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:[:1,1:],REAL:] is set
2 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
[:2,2:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
[:[:2,2:],REAL:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:[:2,2:],REAL:] is set
TOP-REAL 2 is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL 2) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
3 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
TOP-REAL 3 is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL 3) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
INT.Group is non empty non trivial V47() V91() L9()
addint is Relation-like [:INT,INT:] -defined RAT -valued INT -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:INT,INT:],INT:]
G9(INT,addint) is V91() L9()
the carrier of INT.Group is non empty non trivial V32() set
I[01] is non empty strict TopSpace-like V224() V231() pathwise_connected connected TopStruct
the carrier of I[01] is non empty V195() V196() V197() set
RealSpace is strict V231() MetrStruct
[:I[01],I[01]:] is non empty strict TopSpace-like TopStruct
the carrier of [:I[01],I[01]:] is non empty set
K708() is non empty strict TopSpace-like V231() pathwise_connected V240() connected SubSpace of R^1
the carrier of K708() is non empty V195() V196() V197() set
bool the carrier of K708() is V74() set
bool (bool the carrier of K708()) is set
Tunit_circle 2 is non empty TopSpace-like V241() complex-functions-membered real-functions-membered SubSpace of TOP-REAL 2
0. (TOP-REAL 2) is Relation-like NAT -defined Function-like V32() V39(2) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
the ZeroF of (TOP-REAL 2) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Tcircle ((0. (TOP-REAL 2)),1) is non empty strict TopSpace-like pathwise_connected V241() complex-functions-membered real-functions-membered connected SubSpace of TOP-REAL 2
Sphere ((0. (TOP-REAL 2)),1) is functional non empty closed V214( TOP-REAL 2) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 2)
bool the carrier of (TOP-REAL 2) is V74() set
(TOP-REAL 2) | (Sphere ((0. (TOP-REAL 2)),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 2
the carrier of (Tunit_circle 2) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of K708(), the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of K708(), the carrier of (Tunit_circle 2):] is set
CircleMap is Relation-like the carrier of K708() -defined the carrier of K708() -defined the carrier of (Tunit_circle 2) -valued the carrier of (Tunit_circle 2) -valued Function-like non empty total total quasi_total quasi_total onto continuous Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of K708(), the carrier of (Tunit_circle 2):]
c[10] is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty complex ordinal natural real integer non irrational V32() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued Element of NAT
|[1,0]| is Relation-like NAT -defined NAT -defined Function-like non empty V32() V39(2) V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*1*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,1] is set
{1,1} is non empty V195() V196() V197() V198() V199() V200() set
{1} is non empty V195() V196() V197() V198() V199() V200() set
{{1,1},{1}} is non empty set
{[1,1]} is Relation-like Function-like non empty set
<*0*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,0] is set
{1,0} is non empty V195() V196() V197() V198() V199() V200() set
{{1,0},{1}} is non empty set
{[1,0]} is Relation-like Function-like non empty set
<*1*> ^ <*0*> is Relation-like NAT -defined NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
1 + 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
Topen_unit_circle c[10] is non empty strict TopSpace-like V245( Tunit_circle 2) complex-functions-membered real-functions-membered SubSpace of Tunit_circle 2
the carrier of (Topen_unit_circle c[10]) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
].0,1.[ is non empty V195() V196() V197() open Element of bool REAL
R^1 ].0,1.[ is non empty open V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 ].0,1.[) is non empty strict TopSpace-like V231() V245(K708()) V245(K708()) SubSpace of K708()
the carrier of (K708() | (R^1 ].0,1.[)) is non empty V195() V196() V197() set
[: the carrier of (Topen_unit_circle c[10]), the carrier of (K708() | (R^1 ].0,1.[)):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (Topen_unit_circle c[10]), the carrier of (K708() | (R^1 ].0,1.[)):] is set
c[-10] is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
- 1 is non empty complex real integer ext-real non positive negative Element of REAL
|[(- 1),0]| is Relation-like NAT -defined NAT -defined Function-like non empty V32() V39(2) V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(- 1)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(- 1)] is set
{1,(- 1)} is non empty V195() V196() V197() V198() V199() set
{{1,(- 1)},{1}} is non empty set
{[1,(- 1)]} is Relation-like Function-like non empty set
<*(- 1)*> ^ <*0*> is Relation-like NAT -defined NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Topen_unit_circle c[-10] is non empty strict TopSpace-like V245( Tunit_circle 2) complex-functions-membered real-functions-membered SubSpace of Tunit_circle 2
the carrier of (Topen_unit_circle c[-10]) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
1 / 2 is non empty complex real ext-real positive non negative Element of REAL
2 " is non empty complex real ext-real positive non negative set
1 * (2 ") is non empty complex real ext-real positive non negative set
3 / 2 is non empty complex real ext-real positive non negative Element of REAL
3 * (2 ") is non empty complex real ext-real positive non negative set
].(1 / 2),(3 / 2).[ is non empty V195() V196() V197() open Element of bool REAL
R^1 ].(1 / 2),(3 / 2).[ is non empty open V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 ].(1 / 2),(3 / 2).[) is non empty strict TopSpace-like V231() V245(K708()) V245(K708()) SubSpace of K708()
the carrier of (K708() | (R^1 ].(1 / 2),(3 / 2).[)) is non empty V195() V196() V197() set
[: the carrier of (Topen_unit_circle c[-10]), the carrier of (K708() | (R^1 ].(1 / 2),(3 / 2).[)):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (Topen_unit_circle c[-10]), the carrier of (K708() | (R^1 ].(1 / 2),(3 / 2).[)):] is set
[: the carrier of I[01], the carrier of K708():] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of I[01], the carrier of K708():] is set
R^1 0 is complex real ext-real Element of the carrier of K708()
[: the carrier of I[01], the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of I[01], the carrier of (Tunit_circle 2):] is set
K684((Tunit_circle 2),c[10]) is non empty non trivial V47() V91() L9()
the carrier of K684((Tunit_circle 2),c[10]) is non empty non trivial V32() set
[: the carrier of INT.Group, the carrier of K684((Tunit_circle 2),c[10]):] is Relation-like set
bool [: the carrier of INT.Group, the carrier of K684((Tunit_circle 2),c[10]):] is set
[: the carrier of I[01], the carrier of I[01]:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of I[01], the carrier of I[01]:] is set
bool the carrier of [:I[01],I[01]:] is V74() set
[: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):] is Relation-like set
bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):] is set
the carrier of R^1 is non empty V195() V196() V197() set
[: the carrier of (TOP-REAL 2), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL 2), the carrier of R^1:] is set
K942() is set
{{}} is functional non empty complex-functions-membered set
{{}} * is functional non empty FinSequence-membered M29({{}})
[:({{}} *),{{}}:] is Relation-like set
bool [:({{}} *),{{}}:] is set
PFuncs (({{}} *),{{}}) is set
PI is non empty complex real ext-real positive non negative Element of REAL
PI / 2 is non empty complex real ext-real positive non negative Element of REAL
PI * (2 ") is non empty complex real ext-real positive non negative set
- (PI / 2) is non empty complex real ext-real non positive negative Element of REAL
[.(- (PI / 2)),(PI / 2).] is V195() V196() V197() closed Element of bool REAL
(3 / 2) * PI is non empty complex real ext-real positive non negative Element of REAL
[.(PI / 2),((3 / 2) * PI).] is V195() V196() V197() closed Element of bool REAL
[.(- (PI / 2)),0.] is V195() V196() V197() closed Element of bool REAL
[.0,(PI / 2).] is V195() V196() V197() closed Element of bool REAL
[.(PI / 2),PI.] is V195() V196() V197() closed Element of bool REAL
[.PI,((3 / 2) * PI).] is V195() V196() V197() closed Element of bool REAL
2 * PI is non empty complex real ext-real positive non negative Element of REAL
[.((3 / 2) * PI),(2 * PI).] is V195() V196() V197() closed Element of bool REAL
].(- (PI / 2)),(PI / 2).[ is V195() V196() V197() open Element of bool REAL
].(PI / 2),((3 / 2) * PI).[ is V195() V196() V197() open Element of bool REAL
].(- (PI / 2)),0.[ is V195() V196() V197() open Element of bool REAL
].0,(PI / 2).[ is V195() V196() V197() open Element of bool REAL
].(PI / 2),PI.[ is V195() V196() V197() open Element of bool REAL
].PI,((3 / 2) * PI).[ is V195() V196() V197() open Element of bool REAL
].((3 / 2) * PI),(2 * PI).[ is V195() V196() V197() open Element of bool REAL
[.0,PI.] is V195() V196() V197() closed Element of bool REAL
[.PI,(2 * PI).] is V195() V196() V197() closed Element of bool REAL
].0,PI.[ is V195() V196() V197() open Element of bool REAL
].PI,(2 * PI).[ is V195() V196() V197() open Element of bool REAL
sin is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued continuous Element of bool [:REAL,REAL:]
dom sin is non empty V195() V196() V197() Element of bool REAL
cos is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued continuous Element of bool [:REAL,REAL:]
dom cos is non empty V195() V196() V197() Element of bool REAL
cos 0 is complex real ext-real Element of REAL
sin 0 is complex real ext-real Element of REAL
[#] REAL is V195() V196() V197() closed open Element of bool REAL
R^1 ([#] REAL) is open closed V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 ([#] REAL)) is strict TopSpace-like closed V231() V245(K708()) SubSpace of K708()
[: the carrier of K708(), the carrier of K708():] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of K708(), the carrier of K708():] is set
Ciso is Relation-like the carrier of INT.Group -defined the carrier of K684((Tunit_circle 2),c[10]) -valued Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of INT.Group, the carrier of K684((Tunit_circle 2),c[10]):]
K681((Tunit_circle 2),c[10]) is set
K680((Tunit_circle 2),c[10],c[10]) is set
EqRel ((Tunit_circle 2),c[10]) is Relation-like K681((Tunit_circle 2),c[10]) -defined K681((Tunit_circle 2),c[10]) -valued Element of bool [:K681((Tunit_circle 2),c[10]),K681((Tunit_circle 2),c[10]):]
[:K681((Tunit_circle 2),c[10]),K681((Tunit_circle 2),c[10]):] is Relation-like set
bool [:K681((Tunit_circle 2),c[10]),K681((Tunit_circle 2),c[10]):] is set
EqRel ((Tunit_circle 2),c[10],c[10]) is Relation-like K680((Tunit_circle 2),c[10],c[10]) -defined K680((Tunit_circle 2),c[10],c[10]) -valued Element of bool [:K680((Tunit_circle 2),c[10],c[10]),K680((Tunit_circle 2),c[10],c[10]):]
[:K680((Tunit_circle 2),c[10],c[10]),K680((Tunit_circle 2),c[10],c[10]):] is Relation-like set
bool [:K680((Tunit_circle 2),c[10],c[10]),K680((Tunit_circle 2),c[10],c[10]):] is set
absreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
sqrreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
sqrt 1 is non empty complex real ext-real positive non negative Element of REAL
bool the carrier of R^1 is V74() set
Closed-Interval-TSpace (0,1) is non empty strict TopSpace-like V231() SubSpace of R^1
0. (TOP-REAL 3) is Relation-like NAT -defined Function-like V32() V39(3) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
the ZeroF of (TOP-REAL 3) is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
|[0,0,0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*0*> ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*0*> ^ <*0*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 + 1) + 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
[.0,1.] is V195() V196() V197() closed Element of bool REAL
0[01] is complex real ext-real Element of the carrier of I[01]
1[01] is complex real ext-real Element of the carrier of I[01]
+infty is non empty non real ext-real positive non negative set
-infty is non empty non real ext-real non positive negative set
Seg 3 is non empty V32() V39(3) V195() V196() V197() V198() V199() V200() Element of bool NAT
{ b₁ where b₁ is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT : ( 1 <= b₁ & b₁ <= 3 ) } is set
{1,2,3} is V195() V196() V197() V198() V199() V200() set
1r is complex Element of COMPLEX
Re 1r is complex real ext-real Element of REAL
Im 1r is complex real ext-real Element of REAL
<i> is complex Element of COMPLEX
(0,1) --> (0,1) is Relation-like REAL -defined {0,1} -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:{0,1},REAL:]
{0,1} is non empty V195() V196() V197() V198() V199() V200() set
[:{0,1},REAL:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:{0,1},REAL:] is set
0 .--> 0 is Relation-like REAL -defined {0} -defined RAT -valued INT -valued Function-like one-to-one constant trivial complex-yielding ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing Function-yielding V178() set
{0} is functional non empty V195() V196() V197() V198() V199() V200() complex-functions-membered set
{0} --> 0 is Relation-like non non-empty {0} -defined RAT -valued INT -valued {0} -valued Function-like constant non empty total quasi_total complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() complex-functions-valued Element of bool [:{0},{0}:]
[:{0},{0}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
bool [:{0},{0}:] is set
1 .--> 1 is Relation-like REAL -defined {1} -defined RAT -valued INT -valued Function-like one-to-one constant trivial complex-yielding ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing set
{1} --> 1 is Relation-like non-empty {1} -defined RAT -valued INT -valued {1} -valued Function-like constant non empty total quasi_total complex-yielding ext-real-valued real-valued natural-valued Element of bool [:{1},{1}:]
[:{1},{1}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
bool [:{1},{1}:] is set
(0 .--> 0) +* (1 .--> 1) is Relation-like REAL -defined RAT -valued INT -valued Function-like complex-yielding ext-real-valued real-valued natural-valued set
|.0.| is complex real non irrational ext-real Element of REAL
Re 0 is complex real ext-real Element of REAL
(Re 0) ^2 is complex real ext-real non negative Element of REAL
(Re 0) * (Re 0) is complex real ext-real non negative set
Im 0 is complex real ext-real Element of REAL
(Im 0) ^2 is complex real ext-real non negative Element of REAL
(Im 0) * (Im 0) is complex real ext-real non negative set
((Re 0) ^2) + ((Im 0) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re 0) ^2) + ((Im 0) ^2)) is complex real ext-real non negative Element of REAL
diffreal is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K143() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
id REAL is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total quasi_total complex-yielding ext-real-valued real-valued increasing non-decreasing Element of bool [:REAL,REAL:]
K141() is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K738(REAL,K143(),(id REAL),K141()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
IRRAT is non empty V195() V196() V197() Element of bool REAL
<i> is complex set
Euclid 2 is non empty strict Reflexive discerning symmetric triangle complex-functions-membered real-functions-membered MetrStruct
REAL 2 is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered M29( REAL )
K513(2,REAL) is functional non empty FinSequence-membered M29( REAL )
Pitag_dist 2 is Relation-like [:(REAL 2),(REAL 2):] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:(REAL 2),(REAL 2):],REAL:]
[:(REAL 2),(REAL 2):] is Relation-like set
[:[:(REAL 2),(REAL 2):],REAL:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:[:(REAL 2),(REAL 2):],REAL:] is set
MetrStruct(# (REAL 2),(Pitag_dist 2) #) is strict MetrStruct
TopSpaceMetr (Euclid 2) is non empty strict TopSpace-like TopStruct
the carrier of (TopSpaceMetr (Euclid 2)) is non empty set
[: the carrier of (TopSpaceMetr (Euclid 2)), the carrier of (TopSpaceMetr (Euclid 2)):] is Relation-like set
bool [: the carrier of (TopSpaceMetr (Euclid 2)), the carrier of (TopSpaceMetr (Euclid 2)):] is set
0c is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty complex V32() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued Element of COMPLEX
cpx2euc 0c is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Re 0c is complex real ext-real Element of REAL
Im 0c is complex real ext-real Element of REAL
|[(Re 0c),(Im 0c)]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(Re 0c)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Re 0c)] is set
{1,(Re 0c)} is non empty V195() V196() V197() set
{{1,(Re 0c)},{1}} is non empty set
{[1,(Re 0c)]} is Relation-like Function-like non empty set
<*(Im 0c)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Im 0c)] is set
{1,(Im 0c)} is non empty V195() V196() V197() set
{{1,(Im 0c)},{1}} is non empty set
{[1,(Im 0c)]} is Relation-like Function-like non empty set
<*(Re 0c)*> ^ <*(Im 0c)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
euc2cpx (0. (TOP-REAL 2)) is complex Element of COMPLEX
(0. (TOP-REAL 2)) `1 is complex real ext-real Element of REAL
K382((0. (TOP-REAL 2)),1) is complex real ext-real Element of REAL
(0. (TOP-REAL 2)) `2 is complex real ext-real Element of REAL
K382((0. (TOP-REAL 2)),2) is complex real ext-real Element of REAL
((0. (TOP-REAL 2)) `2) * <i> is complex set
((0. (TOP-REAL 2)) `1) + (((0. (TOP-REAL 2)) `2) * <i>) is complex set
[: the carrier of R^1, the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of R^1, the carrier of R^1:] is set
|[0,0]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
[.0,1.[ is V195() V196() V197() Element of bool REAL
R^1 [.0,1.[ is V195() V196() V197() Element of bool the carrier of K708()
R^1 | (R^1 ].0,1.[) is non empty strict TopSpace-like V231() V245( R^1 ) V245(K708()) SubSpace of R^1
[: the carrier of I[01], the carrier of I[01]:] is Relation-like complex-yielding ext-real-valued real-valued set
TC2 is complex real ext-real set
1 / 2 is non empty complex real ext-real positive non negative Element of COMPLEX
TC2 is complex real ext-real set
TC2 + (1 / 2) is complex real ext-real set
Q is complex ordinal natural real integer ext-real non negative set
Q + (1 / 2) is non empty complex real ext-real positive non negative set
(1 / 2) + (1 / 2) is non empty complex real ext-real positive non negative Element of COMPLEX
TC2 is complex real ext-real Element of IRRAT
TC2 is complex real ext-real set
TC2 ^2 is complex real ext-real non negative set
TC2 * TC2 is complex real ext-real non negative set
TC3 is complex real ext-real set
TC3 ^2 is complex real ext-real non negative set
TC3 * TC3 is complex real ext-real non negative set
- TC3 is complex real ext-real set
- (- TC3) is complex real ext-real set
- Q is complex real integer ext-real non positive set
TC2 is complex real ext-real set
frac TC2 is complex real ext-real Element of REAL
[\TC2/] is complex real integer ext-real set
TC2 - [\TC2/] is complex real ext-real set
- [\TC2/] is complex real integer ext-real set
TC2 + (- [\TC2/]) is complex real ext-real set
TC3 is complex real ext-real set
frac TC3 is complex real ext-real Element of REAL
[\TC3/] is complex real integer ext-real set
TC3 - [\TC3/] is complex real ext-real set
- [\TC3/] is complex real integer ext-real set
TC3 + (- [\TC3/]) is complex real ext-real set
TC2 - TC3 is complex real ext-real set
- TC3 is complex real ext-real set
TC2 + (- TC3) is complex real ext-real set
frac (TC2 - TC3) is complex real ext-real Element of REAL
[\(TC2 - TC3)/] is complex real integer ext-real set
(TC2 - TC3) - [\(TC2 - TC3)/] is complex real ext-real set
- [\(TC2 - TC3)/] is complex real integer ext-real set
(TC2 - TC3) + (- [\(TC2 - TC3)/]) is complex real ext-real set
(frac TC2) - (frac TC3) is complex real ext-real Element of REAL
- (frac TC3) is complex real ext-real set
(frac TC2) + (- (frac TC3)) is complex real ext-real set
(TC2 - TC3) - (frac TC2) is complex real ext-real Element of REAL
- (frac TC2) is complex real ext-real set
(TC2 - TC3) + (- (frac TC2)) is complex real ext-real set
((TC2 - TC3) - (frac TC2)) + (frac TC3) is complex real ext-real Element of REAL
TC2 - (frac TC2) is complex real ext-real Element of REAL
TC2 + (- (frac TC2)) is complex real ext-real set
TC3 - (frac TC3) is complex real ext-real Element of REAL
TC3 + (- (frac TC3)) is complex real ext-real set
(TC2 - (frac TC2)) - (TC3 - (frac TC3)) is complex real ext-real Element of REAL
- (TC3 - (frac TC3)) is complex real ext-real set
(TC2 - (frac TC2)) + (- (TC3 - (frac TC3))) is complex real ext-real set
- (frac TC2) is complex real ext-real Element of REAL
(- (frac TC2)) + (frac TC3) is complex real ext-real Element of REAL
(TC2 - TC3) + ((- (frac TC2)) + (frac TC3)) is complex real ext-real Element of REAL
- (frac TC3) is complex real ext-real Element of REAL
(- (frac TC3)) + (frac TC3) is complex real ext-real Element of REAL
(TC2 - TC3) + Q is complex real ext-real set
(TC2 - TC3) - ((frac TC2) - (frac TC3)) is complex real ext-real Element of REAL
- ((frac TC2) - (frac TC3)) is complex real ext-real set
(TC2 - TC3) + (- ((frac TC2) - (frac TC3))) is complex real ext-real set
1 + (frac TC3) is complex real ext-real Element of REAL
(frac TC2) + Q is complex real ext-real Element of REAL
(TC2 - TC3) - 1 is complex real ext-real Element of REAL
- 1 is non empty complex real integer ext-real non positive negative set
(TC2 - TC3) + (- 1) is complex real ext-real set
TC2 is complex real ext-real set
frac TC2 is complex real ext-real Element of REAL
[\TC2/] is complex real integer ext-real set
TC2 - [\TC2/] is complex real ext-real set
- [\TC2/] is complex real integer ext-real set
TC2 + (- [\TC2/]) is complex real ext-real set
TC3 is complex real ext-real set
frac TC3 is complex real ext-real Element of REAL
[\TC3/] is complex real integer ext-real set
TC3 - [\TC3/] is complex real ext-real set
- [\TC3/] is complex real integer ext-real set
TC3 + (- [\TC3/]) is complex real ext-real set
TC2 - TC3 is complex real ext-real set
- TC3 is complex real ext-real set
TC2 + (- TC3) is complex real ext-real set
frac (TC2 - TC3) is complex real ext-real Element of REAL
[\(TC2 - TC3)/] is complex real integer ext-real set
(TC2 - TC3) - [\(TC2 - TC3)/] is complex real ext-real set
- [\(TC2 - TC3)/] is complex real integer ext-real set
(TC2 - TC3) + (- [\(TC2 - TC3)/]) is complex real ext-real set
(frac TC2) - (frac TC3) is complex real ext-real Element of REAL
- (frac TC3) is complex real ext-real set
(frac TC2) + (- (frac TC3)) is complex real ext-real set
((frac TC2) - (frac TC3)) + 1 is complex real ext-real Element of REAL
(TC2 - TC3) - (frac TC2) is complex real ext-real Element of REAL
- (frac TC2) is complex real ext-real set
(TC2 - TC3) + (- (frac TC2)) is complex real ext-real set
((TC2 - TC3) - (frac TC2)) + (frac TC3) is complex real ext-real Element of REAL
(((TC2 - TC3) - (frac TC2)) + (frac TC3)) - 1 is complex real ext-real Element of REAL
- 1 is non empty complex real integer ext-real non positive negative set
(((TC2 - TC3) - (frac TC2)) + (frac TC3)) + (- 1) is complex real ext-real set
TC2 - (frac TC2) is complex real ext-real Element of REAL
TC2 + (- (frac TC2)) is complex real ext-real set
TC3 - (frac TC3) is complex real ext-real Element of REAL
TC3 + (- (frac TC3)) is complex real ext-real set
(TC2 - (frac TC2)) - (TC3 - (frac TC3)) is complex real ext-real Element of REAL
- (TC3 - (frac TC3)) is complex real ext-real set
(TC2 - (frac TC2)) + (- (TC3 - (frac TC3))) is complex real ext-real set
((TC2 - (frac TC2)) - (TC3 - (frac TC3))) - 1 is complex real ext-real Element of REAL
((TC2 - (frac TC2)) - (TC3 - (frac TC3))) + (- 1) is complex real ext-real set
- (frac TC2) is complex real ext-real Element of REAL
(- (frac TC2)) + (frac TC3) is complex real ext-real Element of REAL
((- (frac TC2)) + (frac TC3)) - 1 is complex real ext-real Element of REAL
((- (frac TC2)) + (frac TC3)) + (- 1) is complex real ext-real set
(TC2 - TC3) + (((- (frac TC2)) + (frac TC3)) - 1) is complex real ext-real Element of REAL
1 - 1 is complex real integer ext-real Element of REAL
1 + (- 1) is complex real integer ext-real set
(frac TC3) - 1 is complex real ext-real Element of REAL
(frac TC3) + (- 1) is complex real ext-real set
((frac TC3) - 1) - (frac TC2) is complex real ext-real Element of REAL
((frac TC3) - 1) + (- (frac TC2)) is complex real ext-real set
(frac TC2) - (frac TC2) is complex real ext-real Element of REAL
(frac TC2) + (- (frac TC2)) is complex real ext-real set
(TC2 - TC3) + Q is complex real ext-real set
(TC2 - TC3) - 1 is complex real ext-real Element of REAL
(TC2 - TC3) + (- 1) is complex real ext-real set
((TC2 - TC3) - 1) - ((frac TC2) - (frac TC3)) is complex real ext-real Element of REAL
- ((frac TC2) - (frac TC3)) is complex real ext-real set
((TC2 - TC3) - 1) + (- ((frac TC2) - (frac TC3))) is complex real ext-real set
((TC2 - TC3) - 1) - Q is complex real ext-real Element of REAL
- Q is complex real integer ext-real non positive set
((TC2 - TC3) - 1) + (- Q) is complex real ext-real set
TC2 is complex real ext-real set
frac TC2 is complex real ext-real Element of REAL
[\TC2/] is complex real integer ext-real set
TC2 - [\TC2/] is complex real ext-real set
- [\TC2/] is complex real integer ext-real set
TC2 + (- [\TC2/]) is complex real ext-real set
TC3 is complex real ext-real set
TC2 - TC3 is complex real ext-real set
- TC3 is complex real ext-real set
TC2 + (- TC3) is complex real ext-real set
frac (TC2 - TC3) is complex real ext-real Element of REAL
[\(TC2 - TC3)/] is complex real integer ext-real set
(TC2 - TC3) - [\(TC2 - TC3)/] is complex real ext-real set
- [\(TC2 - TC3)/] is complex real integer ext-real set
(TC2 - TC3) + (- [\(TC2 - TC3)/]) is complex real ext-real set
frac TC3 is complex real ext-real Element of REAL
[\TC3/] is complex real integer ext-real set
TC3 - [\TC3/] is complex real ext-real set
- [\TC3/] is complex real integer ext-real set
TC3 + (- [\TC3/]) is complex real ext-real set
(frac TC2) - (frac TC3) is complex real ext-real Element of REAL
- (frac TC3) is complex real ext-real set
(frac TC2) + (- (frac TC3)) is complex real ext-real set
((frac TC2) - (frac TC3)) + Q is complex real ext-real Element of REAL
((frac TC2) - (frac TC3)) + 1 is complex real ext-real Element of REAL
TC2 is complex real ext-real set
sin TC2 is complex real ext-real set
TC2 / (2 * PI) is complex real ext-real Element of COMPLEX
(2 * PI) " is non empty complex real ext-real positive non negative set
TC2 * ((2 * PI) ") is complex real ext-real set
[\(TC2 / (2 * PI))/] is complex real integer ext-real set
(2 * PI) * [\(TC2 / (2 * PI))/] is complex real ext-real Element of REAL
PI + ((2 * PI) * [\(TC2 / (2 * PI))/]) is complex real ext-real Element of REAL
- [\(TC2 / (2 * PI))/] is complex real integer ext-real set
(2 * PI) * (- [\(TC2 / (2 * PI))/]) is complex real ext-real Element of REAL
((2 * PI) * (- [\(TC2 / (2 * PI))/])) + TC2 is complex real ext-real Element of REAL
QQ is complex real ext-real set
sin QQ is complex real ext-real set
PI / 2 is non empty complex real ext-real positive non negative Element of COMPLEX
3 * PI is non empty complex real ext-real positive non negative Element of REAL
(3 * PI) / 2 is non empty complex real ext-real positive non negative Element of COMPLEX
(3 * PI) * (2 ") is non empty complex real ext-real positive non negative set
TC2 is complex real ext-real set
cos TC2 is complex real ext-real set
TC2 / (2 * PI) is complex real ext-real Element of COMPLEX
(2 * PI) " is non empty complex real ext-real positive non negative set
TC2 * ((2 * PI) ") is complex real ext-real set
[\(TC2 / (2 * PI))/] is complex real integer ext-real set
(2 * PI) * [\(TC2 / (2 * PI))/] is complex real ext-real Element of REAL
(PI / 2) + ((2 * PI) * [\(TC2 / (2 * PI))/]) is complex real ext-real Element of REAL
((3 * PI) / 2) + ((2 * PI) * [\(TC2 / (2 * PI))/]) is complex real ext-real Element of REAL
- [\(TC2 / (2 * PI))/] is complex real integer ext-real set
(2 * PI) * (- [\(TC2 / (2 * PI))/]) is complex real ext-real Element of REAL
((2 * PI) * (- [\(TC2 / (2 * PI))/])) + TC2 is complex real ext-real Element of REAL
QQ is complex real ext-real set
cos QQ is complex real ext-real set
TC2 is complex real ext-real set
sin TC2 is complex real ext-real set
TC2 / (2 * PI) is complex real ext-real Element of COMPLEX
(2 * PI) " is non empty complex real ext-real positive non negative set
TC2 * ((2 * PI) ") is complex real ext-real set
[\(TC2 / (2 * PI))/] is complex real integer ext-real set
(2 * PI) * [\(TC2 / (2 * PI))/] is complex real ext-real Element of REAL
PI + ((2 * PI) * [\(TC2 / (2 * PI))/]) is complex real ext-real Element of REAL
2 * [\(TC2 / (2 * PI))/] is complex real integer ext-real even Element of REAL
PI * (2 * [\(TC2 / (2 * PI))/]) is complex real ext-real Element of REAL
1 + (2 * [\(TC2 / (2 * PI))/]) is non empty complex real integer ext-real non even Element of REAL
PI * (1 + (2 * [\(TC2 / (2 * PI))/])) is non empty complex real ext-real Element of REAL
TC2 is complex real ext-real set
cos TC2 is complex real ext-real set
TC2 / (2 * PI) is complex real ext-real Element of COMPLEX
(2 * PI) " is non empty complex real ext-real positive non negative set
TC2 * ((2 * PI) ") is complex real ext-real set
[\(TC2 / (2 * PI))/] is complex real integer ext-real set
(2 * PI) * [\(TC2 / (2 * PI))/] is complex real ext-real Element of REAL
(PI / 2) + ((2 * PI) * [\(TC2 / (2 * PI))/]) is complex real ext-real Element of REAL
((3 * PI) / 2) + ((2 * PI) * [\(TC2 / (2 * PI))/]) is complex real ext-real Element of REAL
2 * [\(TC2 / (2 * PI))/] is complex real integer ext-real even Element of REAL
PI * (2 * [\(TC2 / (2 * PI))/]) is complex real ext-real Element of REAL
(PI / 2) + (PI * (2 * [\(TC2 / (2 * PI))/])) is complex real ext-real Element of REAL
1 + (2 * [\(TC2 / (2 * PI))/]) is non empty complex real integer ext-real non even Element of REAL
PI * (1 + (2 * [\(TC2 / (2 * PI))/])) is non empty complex real ext-real Element of REAL
(PI / 2) + (PI * (1 + (2 * [\(TC2 / (2 * PI))/]))) is complex real ext-real Element of REAL
TC2 is complex real ext-real set
TC3 is complex real ext-real set
TC2 - TC3 is complex real ext-real set
- TC3 is complex real ext-real set
TC2 + (- TC3) is complex real ext-real set
(TC2 - TC3) / 2 is complex real ext-real Element of COMPLEX
(TC2 - TC3) * (2 ") is complex real ext-real set
sin ((TC2 - TC3) / 2) is complex real ext-real set
QQ is complex real integer ext-real set
PI * QQ is complex real ext-real Element of REAL
(2 * PI) * QQ is complex real ext-real Element of REAL
TC3 + ((2 * PI) * QQ) is complex real ext-real Element of REAL
TC2 is complex real ext-real set
sin TC2 is complex real ext-real set
TC3 is complex real ext-real set
sin TC3 is complex real ext-real set
PI - TC3 is complex real ext-real Element of REAL
- TC3 is complex real ext-real set
PI + (- TC3) is complex real ext-real set
(sin TC2) - (sin TC3) is complex real ext-real set
- (sin TC3) is complex real ext-real set
(sin TC2) + (- (sin TC3)) is complex real ext-real set
TC2 + TC3 is complex real ext-real set
(TC2 + TC3) / 2 is complex real ext-real Element of COMPLEX
(TC2 + TC3) * (2 ") is complex real ext-real set
cos ((TC2 + TC3) / 2) is complex real ext-real set
TC2 - TC3 is complex real ext-real set
TC2 + (- TC3) is complex real ext-real set
(TC2 - TC3) / 2 is complex real ext-real Element of COMPLEX
(TC2 - TC3) * (2 ") is complex real ext-real set
sin ((TC2 - TC3) / 2) is complex real ext-real set
(cos ((TC2 + TC3) / 2)) * (sin ((TC2 - TC3) / 2)) is complex real ext-real set
2 * ((cos ((TC2 + TC3) / 2)) * (sin ((TC2 - TC3) / 2))) is complex real ext-real Element of REAL
QQ is complex real integer ext-real set
PI * QQ is complex real ext-real Element of REAL
(PI / 2) + (PI * QQ) is complex real ext-real Element of REAL
(2 * PI) * QQ is complex real ext-real Element of REAL
(PI - TC3) + ((2 * PI) * QQ) is complex real ext-real Element of REAL
TC2 is complex real ext-real set
cos TC2 is complex real ext-real set
TC3 is complex real ext-real set
cos TC3 is complex real ext-real set
- TC3 is complex real ext-real set
(cos TC2) - (cos TC3) is complex real ext-real set
- (cos TC3) is complex real ext-real set
(cos TC2) + (- (cos TC3)) is complex real ext-real set
TC2 + TC3 is complex real ext-real set
(TC2 + TC3) / 2 is complex real ext-real Element of COMPLEX
(TC2 + TC3) * (2 ") is complex real ext-real set
sin ((TC2 + TC3) / 2) is complex real ext-real set
TC2 - TC3 is complex real ext-real set
TC2 + (- TC3) is complex real ext-real set
(TC2 - TC3) / 2 is complex real ext-real Element of COMPLEX
(TC2 - TC3) * (2 ") is complex real ext-real set
sin ((TC2 - TC3) / 2) is complex real ext-real set
(sin ((TC2 + TC3) / 2)) * (sin ((TC2 - TC3) / 2)) is complex real ext-real set
2 * ((sin ((TC2 + TC3) / 2)) * (sin ((TC2 - TC3) / 2))) is complex real ext-real Element of REAL
- (2 * ((sin ((TC2 + TC3) / 2)) * (sin ((TC2 - TC3) / 2)))) is complex real ext-real Element of REAL
QQ is complex real integer ext-real set
PI * QQ is complex real ext-real Element of REAL
(TC2 + TC3) - TC3 is complex real ext-real set
(TC2 + TC3) + (- TC3) is complex real ext-real set
(2 * PI) * QQ is complex real ext-real Element of REAL
((2 * PI) * QQ) - TC3 is complex real ext-real Element of REAL
((2 * PI) * QQ) + (- TC3) is complex real ext-real set
TC2 is complex real ext-real set
sin TC2 is complex real ext-real set
cos TC2 is complex real ext-real set
TC3 is complex real ext-real set
sin TC3 is complex real ext-real set
cos TC3 is complex real ext-real set
PI - TC3 is complex real ext-real Element of REAL
- TC3 is complex real ext-real set
PI + (- TC3) is complex real ext-real set
QQ is complex real integer ext-real set
(2 * PI) * QQ is complex real ext-real Element of REAL
TC3 + ((2 * PI) * QQ) is complex real ext-real Element of REAL
(PI - TC3) + ((2 * PI) * QQ) is complex real ext-real Element of REAL
RR is complex real integer ext-real set
(2 * PI) * RR is complex real ext-real Element of REAL
TC3 + ((2 * PI) * RR) is complex real ext-real Element of REAL
(- TC3) + ((2 * PI) * RR) is complex real ext-real Element of REAL
PI / PI is non empty complex real ext-real positive non negative Element of COMPLEX
PI " is non empty complex real ext-real positive non negative set
PI * (PI ") is non empty complex real ext-real positive non negative set
RR - QQ is complex real integer ext-real set
- QQ is complex real integer ext-real set
RR + (- QQ) is complex real integer ext-real set
2 * (RR - QQ) is complex real integer ext-real even Element of REAL
PI * (2 * (RR - QQ)) is complex real ext-real Element of REAL
(PI * (2 * (RR - QQ))) / PI is complex real ext-real Element of COMPLEX
(PI * (2 * (RR - QQ))) * (PI ") is complex real ext-real set
TC2 is complex real integer ext-real set
(2 * PI) * TC2 is complex real ext-real Element of REAL
TC3 is complex set
|.TC3.| is complex real ext-real Element of REAL
Re TC3 is complex real ext-real Element of REAL
(Re TC3) ^2 is complex real ext-real non negative Element of REAL
(Re TC3) * (Re TC3) is complex real ext-real non negative set
Im TC3 is complex real ext-real Element of REAL
(Im TC3) ^2 is complex real ext-real non negative Element of REAL
(Im TC3) * (Im TC3) is complex real ext-real non negative set
((Re TC3) ^2) + ((Im TC3) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re TC3) ^2) + ((Im TC3) ^2)) is complex real ext-real non negative Element of REAL
Arg TC3 is complex real ext-real Element of REAL
QQ is complex set
|.QQ.| is complex real ext-real Element of REAL
Re QQ is complex real ext-real Element of REAL
(Re QQ) ^2 is complex real ext-real non negative Element of REAL
(Re QQ) * (Re QQ) is complex real ext-real non negative set
Im QQ is complex real ext-real Element of REAL
(Im QQ) ^2 is complex real ext-real non negative Element of REAL
(Im QQ) * (Im QQ) is complex real ext-real non negative set
((Re QQ) ^2) + ((Im QQ) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re QQ) ^2) + ((Im QQ) ^2)) is complex real ext-real non negative Element of REAL
Arg QQ is complex real ext-real Element of REAL
(Arg QQ) + ((2 * PI) * TC2) is complex real ext-real Element of REAL
cos (Arg QQ) is complex real ext-real Element of REAL
cos ((Arg QQ) + ((2 * PI) * TC2)) is complex real ext-real Element of REAL
sin (Arg QQ) is complex real ext-real Element of REAL
sin ((Arg QQ) + ((2 * PI) * TC2)) is complex real ext-real Element of REAL
cos (Arg TC3) is complex real ext-real Element of REAL
|.TC3.| * (cos (Arg TC3)) is complex real ext-real Element of REAL
sin (Arg TC3) is complex real ext-real Element of REAL
|.TC3.| * (sin (Arg TC3)) is complex real ext-real Element of REAL
(|.TC3.| * (sin (Arg TC3))) * <i> is complex set
(|.TC3.| * (cos (Arg TC3))) + ((|.TC3.| * (sin (Arg TC3))) * <i>) is complex set
|.QQ.| * (cos (Arg QQ)) is complex real ext-real Element of REAL
|.QQ.| * (sin (Arg QQ)) is complex real ext-real Element of REAL
(|.QQ.| * (sin (Arg QQ))) * <i> is complex set
(|.QQ.| * (cos (Arg QQ))) + ((|.QQ.| * (sin (Arg QQ))) * <i>) is complex set
TC2 is Relation-like Function-like one-to-one complex-yielding set
TC3 is complex set
TC3 + TC2 is Relation-like Function-like complex-yielding set
QQ is set
dom (TC3 + TC2) is set
(TC3 + TC2) . QQ is complex set
RR is set
(TC3 + TC2) . RR is complex set
dom TC2 is set
TC2 . QQ is complex set
(TC2 . QQ) + TC3 is complex set
TC2 . RR is complex set
(TC2 . RR) + TC3 is complex set
TC2 is Relation-like Function-like one-to-one complex-yielding set
TC3 is complex set
TC2 - TC3 is Relation-like Function-like complex-yielding set
- TC3 is complex set
(- TC3) + TC2 is Relation-like Function-like one-to-one complex-yielding set
TC2 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
- TC2 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
- 1 is non empty complex real integer ext-real non positive negative set
(- 1) * TC2 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
len (- TC2) is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT
len TC2 is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT
Seg (len (- TC2)) is V32() V39( len (- TC2)) V195() V196() V197() V198() V199() V200() Element of bool NAT
{ b₁ where b₁ is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT : ( 1 <= b₁ & b₁ <= len (- TC2) ) } is set
dom (- TC2) is V195() V196() V197() V198() V199() V200() Element of bool NAT
dom TC2 is V195() V196() V197() V198() V199() V200() Element of bool NAT
Seg (len TC2) is V32() V39( len TC2) V195() V196() V197() V198() V199() V200() Element of bool NAT
{ b₁ where b₁ is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT : ( 1 <= b₁ & b₁ <= len TC2 ) } is set
TC2 is complex ordinal natural real integer ext-real non negative set
0* TC2 is Relation-like NAT -defined REAL -valued Function-like V32() V39(TC2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of REAL TC2
REAL TC2 is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered M29( REAL )
K513(TC2,REAL) is functional non empty FinSequence-membered M29( REAL )
K514(REAL,TC2,0) is Relation-like empty-yielding NAT -defined REAL -valued Function-like V32() V39(TC2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of K513(TC2,REAL)
- (0* TC2) is Relation-like NAT -defined REAL -valued Function-like V32() V39(TC2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of REAL TC2
- 1 is non empty complex real integer ext-real non positive negative set
(- 1) * (0* TC2) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(0* TC2) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(0* TC2) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
len (0* TC2) is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT
len (- (0* TC2)) is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT
RR is complex ordinal natural real integer ext-real non negative set
(- (0* TC2)) . RR is complex real ext-real set
(0* TC2) . RR is complex real ext-real set
- Q is complex real integer ext-real non positive set
TC2 is complex ordinal natural real integer ext-real non negative set
0* TC2 is Relation-like NAT -defined REAL -valued Function-like V32() V39(TC2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of REAL TC2
REAL TC2 is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered M29( REAL )
K513(TC2,REAL) is functional non empty FinSequence-membered M29( REAL )
K514(REAL,TC2,0) is Relation-like empty-yielding NAT -defined REAL -valued Function-like V32() V39(TC2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of K513(TC2,REAL)
TC3 is Relation-like Function-like complex-yielding set
- TC3 is Relation-like Function-like complex-yielding set
- 1 is non empty complex real integer ext-real non positive negative set
(- 1) * TC3 is Relation-like Function-like complex-yielding set
- (- TC3) is Relation-like Function-like complex-yielding set
(- 1) * (- TC3) is Relation-like Function-like complex-yielding set
- (0* TC2) is Relation-like NAT -defined REAL -valued Function-like V32() V39(TC2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of REAL TC2
(- 1) * (0* TC2) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(0* TC2) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(0* TC2) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
TC2 is complex real ext-real set
TC2 ^2 is complex real ext-real non negative set
TC2 * TC2 is complex real ext-real non negative set
TC3 is complex real ext-real set
TC3 ^2 is complex real ext-real non negative set
TC3 * TC3 is complex real ext-real non negative set
QQ is complex real ext-real set
<*TC2,TC3,QQ*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*TC2*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,TC2] is set
{1,TC2} is non empty V195() V196() V197() set
{{1,TC2},{1}} is non empty set
{[1,TC2]} is Relation-like Function-like non empty set
<*TC3*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,TC3] is set
{1,TC3} is non empty V195() V196() V197() set
{{1,TC3},{1}} is non empty set
{[1,TC3]} is Relation-like Function-like non empty set
<*TC2*> ^ <*TC3*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*QQ*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,QQ] is set
{1,QQ} is non empty V195() V196() V197() set
{{1,QQ},{1}} is non empty set
{[1,QQ]} is Relation-like Function-like non empty set
(<*TC2*> ^ <*TC3*>) ^ <*QQ*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
sqr <*TC2,TC3,QQ*> is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (<*TC2,TC3,QQ*>,<*TC2,TC3,QQ*>) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (<*TC2,TC3,QQ*>,<*TC2,TC3,QQ*>) is Relation-like Function-like set
K254(<*TC2,TC3,QQ*>,<*TC2,TC3,QQ*>) is Relation-like Function-like set
K254(<*TC2,TC3,QQ*>,<*TC2,TC3,QQ*>) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
<*TC2,TC3,QQ*> (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
QQ ^2 is complex real ext-real non negative set
QQ * QQ is complex real ext-real non negative set
<*(TC2 ^2),(TC3 ^2),(QQ ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(TC2 ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(TC2 ^2)] is set
{1,(TC2 ^2)} is non empty V195() V196() V197() set
{{1,(TC2 ^2)},{1}} is non empty set
{[1,(TC2 ^2)]} is Relation-like Function-like non empty set
<*(TC3 ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(TC3 ^2)] is set
{1,(TC3 ^2)} is non empty V195() V196() V197() set
{{1,(TC3 ^2)},{1}} is non empty set
{[1,(TC3 ^2)]} is Relation-like Function-like non empty set
<*(TC2 ^2)*> ^ <*(TC3 ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(QQ ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(QQ ^2)] is set
{1,(QQ ^2)} is non empty V195() V196() V197() set
{{1,(QQ ^2)},{1}} is non empty set
{[1,(QQ ^2)]} is Relation-like Function-like non empty set
(<*(TC2 ^2)*> ^ <*(TC3 ^2)*>) ^ <*(QQ ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*TC2,TC3*> is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
sqr <*TC2,TC3*> is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (<*TC2,TC3*>,<*TC2,TC3*>) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (<*TC2,TC3*>,<*TC2,TC3*>) is Relation-like Function-like set
K254(<*TC2,TC3*>,<*TC2,TC3*>) is Relation-like Function-like set
K254(<*TC2,TC3*>,<*TC2,TC3*>) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
<*TC2,TC3*> (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
sqr <*QQ*> is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (<*QQ*>,<*QQ*>) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (<*QQ*>,<*QQ*>) is Relation-like Function-like set
K254(<*QQ*>,<*QQ*>) is Relation-like Function-like set
K254(<*QQ*>,<*QQ*>) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
<*QQ*> (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(sqr <*TC2,TC3*>) ^ (sqr <*QQ*>) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
<*(TC2 ^2),(TC3 ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(TC2 ^2),(TC3 ^2)*> ^ (sqr <*QQ*>) is Relation-like NAT -defined Function-like non empty V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
TC2 is complex real ext-real set
TC2 ^2 is complex real ext-real non negative set
TC2 * TC2 is complex real ext-real non negative set
TC3 is complex real ext-real set
TC3 ^2 is complex real ext-real non negative set
TC3 * TC3 is complex real ext-real non negative set
(TC2 ^2) + (TC3 ^2) is complex real ext-real non negative set
QQ is complex real ext-real set
<*TC2,TC3,QQ*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*TC2*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,TC2] is set
{1,TC2} is non empty V195() V196() V197() set
{{1,TC2},{1}} is non empty set
{[1,TC2]} is Relation-like Function-like non empty set
<*TC3*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,TC3] is set
{1,TC3} is non empty V195() V196() V197() set
{{1,TC3},{1}} is non empty set
{[1,TC3]} is Relation-like Function-like non empty set
<*TC2*> ^ <*TC3*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*QQ*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,QQ] is set
{1,QQ} is non empty V195() V196() V197() set
{{1,QQ},{1}} is non empty set
{[1,QQ]} is Relation-like Function-like non empty set
(<*TC2*> ^ <*TC3*>) ^ <*QQ*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
sqr <*TC2,TC3,QQ*> is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (<*TC2,TC3,QQ*>,<*TC2,TC3,QQ*>) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (<*TC2,TC3,QQ*>,<*TC2,TC3,QQ*>) is Relation-like Function-like set
K254(<*TC2,TC3,QQ*>,<*TC2,TC3,QQ*>) is Relation-like Function-like set
K254(<*TC2,TC3,QQ*>,<*TC2,TC3,QQ*>) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
<*TC2,TC3,QQ*> (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr <*TC2,TC3,QQ*>) is complex real ext-real Element of REAL
QQ ^2 is complex real ext-real non negative set
QQ * QQ is complex real ext-real non negative set
((TC2 ^2) + (TC3 ^2)) + (QQ ^2) is complex real ext-real non negative set
<*(TC2 ^2),(TC3 ^2),(QQ ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(TC2 ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(TC2 ^2)] is set
{1,(TC2 ^2)} is non empty V195() V196() V197() set
{{1,(TC2 ^2)},{1}} is non empty set
{[1,(TC2 ^2)]} is Relation-like Function-like non empty set
<*(TC3 ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(TC3 ^2)] is set
{1,(TC3 ^2)} is non empty V195() V196() V197() set
{{1,(TC3 ^2)},{1}} is non empty set
{[1,(TC3 ^2)]} is Relation-like Function-like non empty set
<*(TC2 ^2)*> ^ <*(TC3 ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(QQ ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(QQ ^2)] is set
{1,(QQ ^2)} is non empty V195() V196() V197() set
{{1,(QQ ^2)},{1}} is non empty set
{[1,(QQ ^2)]} is Relation-like Function-like non empty set
(<*(TC2 ^2)*> ^ <*(TC3 ^2)*>) ^ <*(QQ ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum <*(TC2 ^2),(TC3 ^2),(QQ ^2)*> is complex real ext-real set
TC2 is complex set
TC2 ^2 is complex set
TC2 * TC2 is complex set
TC3 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
TC2 * TC3 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
sqr (TC2 * TC3) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
mlt ((TC2 * TC3),(TC2 * TC3)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
sqr TC3 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
mlt (TC3,TC3) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
(TC2 ^2) * (sqr TC3) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
dom (sqr (TC2 * TC3)) is V195() V196() V197() V198() V199() V200() Element of bool NAT
dom (TC2 * TC3) is V195() V196() V197() V198() V199() V200() Element of bool NAT
dom TC3 is V195() V196() V197() V198() V199() V200() Element of bool NAT
dom ((TC2 ^2) * (sqr TC3)) is V195() V196() V197() V198() V199() V200() Element of bool NAT
dom (sqr TC3) is V195() V196() V197() V198() V199() V200() Element of bool NAT
QQ is set
(sqr (TC2 * TC3)) . QQ is complex set
(TC2 * TC3) . QQ is complex set
((TC2 * TC3) . QQ) ^2 is complex set
((TC2 * TC3) . QQ) * ((TC2 * TC3) . QQ) is complex set
TC3 . QQ is complex set
TC2 * (TC3 . QQ) is complex set
(TC2 * (TC3 . QQ)) ^2 is complex set
(TC2 * (TC3 . QQ)) * (TC2 * (TC3 . QQ)) is complex set
(TC3 . QQ) ^2 is complex set
(TC3 . QQ) * (TC3 . QQ) is complex set
(TC2 ^2) * ((TC3 . QQ) ^2) is complex set
(sqr TC3) . QQ is complex set
(TC2 ^2) * ((sqr TC3) . QQ) is complex set
((TC2 ^2) * (sqr TC3)) . QQ is complex set
TC2 is complex set
TC2 ^2 is complex set
TC2 * TC2 is complex set
TC3 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
TC3 (/) TC2 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
1 / TC2 is complex set
TC2 " is complex set
1 * (TC2 ") is complex set
(1 / TC2) * TC3 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
sqr (TC3 (/) TC2) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
mlt ((TC3 (/) TC2),(TC3 (/) TC2)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
sqr TC3 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
mlt (TC3,TC3) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
(sqr TC3) (/) (TC2 ^2) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
1 / (TC2 ^2) is complex set
(TC2 ^2) " is complex set
1 * ((TC2 ^2) ") is complex set
(1 / (TC2 ^2)) * (sqr TC3) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
1 / TC2 is complex Element of COMPLEX
(1 / TC2) ^2 is complex Element of COMPLEX
(1 / TC2) * (1 / TC2) is complex set
((1 / TC2) ^2) * (sqr TC3) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
1 * 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
(1 * 1) / (TC2 * TC2) is complex Element of COMPLEX
(TC2 * TC2) " is complex set
(1 * 1) * ((TC2 * TC2) ") is complex set
((1 * 1) / (TC2 * TC2)) * (sqr TC3) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding set
TC2 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum TC2 is complex real ext-real set
TC2 (/) (Sum TC2) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
1 / (Sum TC2) is complex real ext-real set
(Sum TC2) " is complex real ext-real set
1 * ((Sum TC2) ") is complex real ext-real set
(1 / (Sum TC2)) * TC2 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 / (Sum TC2)) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] ((1 / (Sum TC2)),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (1 / (Sum TC2)) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 / (Sum TC2))} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 / (Sum TC2))}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 / (Sum TC2))),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 / (Sum TC2))),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
TC2 (#) ((1 / (Sum TC2)) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (TC2 (/) (Sum TC2)) is complex real ext-real set
(Sum TC2) / (Sum TC2) is complex real ext-real non negative Element of COMPLEX
(Sum TC2) * ((Sum TC2) ") is complex real ext-real non negative set
TC2 is set
TC3 is set
RR is set
c100 is set
(TC2,TC3) --> (RR,c100) is Relation-like Function-like set
TC2 .--> RR is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> RR is Relation-like {TC2} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{RR}:]
{RR} is non empty set
[:{TC2},{RR}:] is Relation-like set
bool [:{TC2},{RR}:] is set
TC3 .--> c100 is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> c100 is Relation-like {TC3} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{c100}:]
{c100} is non empty set
[:{TC3},{c100}:] is Relation-like set
bool [:{TC3},{c100}:] is set
(TC2 .--> RR) +* (TC3 .--> c100) is Relation-like Function-like set
QQ is set
c100a is set
QQ .--> c100a is Relation-like {QQ} -defined Function-like one-to-one constant trivial set
{QQ} is non empty set
{QQ} --> c100a is Relation-like {QQ} -defined {c100a} -valued Function-like constant non empty total quasi_total Element of bool [:{QQ},{c100a}:]
{c100a} is non empty set
[:{QQ},{c100a}:] is Relation-like set
bool [:{QQ},{c100a}:] is set
((TC2,TC3) --> (RR,c100)) +* (QQ .--> c100a) is Relation-like Function-like set
TC2 is set
TC3 is set
QQ is set
RR is set
c100 is set
c100a is set
(TC2,TC3,QQ,RR,c100,c100a) is set
(TC2,TC3) --> (RR,c100) is Relation-like Function-like set
TC2 .--> RR is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> RR is Relation-like {TC2} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{RR}:]
{RR} is non empty set
[:{TC2},{RR}:] is Relation-like set
bool [:{TC2},{RR}:] is set
TC3 .--> c100 is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> c100 is Relation-like {TC3} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{c100}:]
{c100} is non empty set
[:{TC3},{c100}:] is Relation-like set
bool [:{TC3},{c100}:] is set
(TC2 .--> RR) +* (TC3 .--> c100) is Relation-like Function-like set
QQ .--> c100a is Relation-like {QQ} -defined Function-like one-to-one constant trivial set
{QQ} is non empty set
{QQ} --> c100a is Relation-like {QQ} -defined {c100a} -valued Function-like constant non empty total quasi_total Element of bool [:{QQ},{c100a}:]
{c100a} is non empty set
[:{QQ},{c100a}:] is Relation-like set
bool [:{QQ},{c100a}:] is set
((TC2,TC3) --> (RR,c100)) +* (QQ .--> c100a) is Relation-like Function-like set
TC2 is set
TC3 is set
c100a is complex set
QQ is set
RR is set
c100 is set
(TC2,TC3,c100a,QQ,RR,c100) is Relation-like Function-like set
(TC2,TC3) --> (QQ,RR) is Relation-like Function-like set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
{QQ} is non empty set
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
TC3 .--> RR is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> RR is Relation-like {TC3} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{RR}:]
{RR} is non empty set
[:{TC3},{RR}:] is Relation-like set
bool [:{TC3},{RR}:] is set
(TC2 .--> QQ) +* (TC3 .--> RR) is Relation-like Function-like set
c100a .--> c100 is Relation-like {c100a} -defined Function-like one-to-one constant trivial set
{c100a} is non empty V195() set
{c100a} --> c100 is Relation-like {c100a} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{c100a},{c100}:]
{c100} is non empty set
[:{c100a},{c100}:] is Relation-like set
bool [:{c100a},{c100}:] is set
((TC2,TC3) --> (QQ,RR)) +* (c100a .--> c100) is Relation-like Function-like set
dom (TC2,TC3,c100a,QQ,RR,c100) is set
{TC2,TC3,c100a} is set
TC2 is set
TC3 is set
QQ is set
RR is set
c100 is set
{QQ,RR,c100} is set
c100a is complex set
(TC2,TC3,c100a,QQ,RR,c100) is Relation-like Function-like set
(TC2,TC3) --> (QQ,RR) is Relation-like Function-like set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
{QQ} is non empty set
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
TC3 .--> RR is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> RR is Relation-like {TC3} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{RR}:]
{RR} is non empty set
[:{TC3},{RR}:] is Relation-like set
bool [:{TC3},{RR}:] is set
(TC2 .--> QQ) +* (TC3 .--> RR) is Relation-like Function-like set
c100a .--> c100 is Relation-like {c100a} -defined Function-like one-to-one constant trivial set
{c100a} is non empty V195() set
{c100a} --> c100 is Relation-like {c100a} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{c100a},{c100}:]
{c100} is non empty set
[:{c100a},{c100}:] is Relation-like set
bool [:{c100a},{c100}:] is set
((TC2,TC3) --> (QQ,RR)) +* (c100a .--> c100) is Relation-like Function-like set
rng (TC2,TC3,c100a,QQ,RR,c100) is set
{QQ,RR} is non empty set
{QQ,RR} \/ {c100} is non empty set
rng ((TC2,TC3) --> (QQ,RR)) is set
rng (c100a .--> c100) is trivial set
(rng ((TC2,TC3) --> (QQ,RR))) \/ (rng (c100a .--> c100)) is set
TC2 is set
TC3 is set
QQ is set
RR is set
(TC2,TC2,TC2,TC3,QQ,RR) is Relation-like Function-like set
(TC2,TC2) --> (TC3,QQ) is Relation-like Function-like set
TC2 .--> TC3 is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> TC3 is Relation-like {TC2} -defined {TC3} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{TC3}:]
{TC3} is non empty set
[:{TC2},{TC3}:] is Relation-like set
bool [:{TC2},{TC3}:] is set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
{QQ} is non empty set
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
(TC2 .--> TC3) +* (TC2 .--> QQ) is Relation-like {TC2} -defined Function-like set
TC2 .--> RR is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> RR is Relation-like {TC2} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{RR}:]
{RR} is non empty set
[:{TC2},{RR}:] is Relation-like set
bool [:{TC2},{RR}:] is set
((TC2,TC2) --> (TC3,QQ)) +* (TC2 .--> RR) is Relation-like Function-like set
(TC2,TC2) --> (QQ,RR) is Relation-like Function-like set
(TC2 .--> QQ) +* (TC2 .--> RR) is Relation-like {TC2} -defined Function-like set
TC2 is set
TC3 is set
QQ is set
RR is set
c100 is set
(TC2,TC2,TC3,QQ,RR,c100) is Relation-like Function-like set
(TC2,TC2) --> (QQ,RR) is Relation-like Function-like set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
{QQ} is non empty set
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
TC2 .--> RR is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> RR is Relation-like {TC2} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{RR}:]
{RR} is non empty set
[:{TC2},{RR}:] is Relation-like set
bool [:{TC2},{RR}:] is set
(TC2 .--> QQ) +* (TC2 .--> RR) is Relation-like {TC2} -defined Function-like set
TC3 .--> c100 is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> c100 is Relation-like {TC3} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{c100}:]
{c100} is non empty set
[:{TC3},{c100}:] is Relation-like set
bool [:{TC3},{c100}:] is set
((TC2,TC2) --> (QQ,RR)) +* (TC3 .--> c100) is Relation-like Function-like set
(TC2,TC3) --> (RR,c100) is Relation-like Function-like set
(TC2 .--> RR) +* (TC3 .--> c100) is Relation-like Function-like set
TC2 is set
TC3 is set
QQ is set
RR is set
c100 is set
(TC2,TC3,TC2,QQ,RR,c100) is Relation-like Function-like set
(TC2,TC3) --> (QQ,RR) is Relation-like Function-like set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
{QQ} is non empty set
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
TC3 .--> RR is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> RR is Relation-like {TC3} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{RR}:]
{RR} is non empty set
[:{TC3},{RR}:] is Relation-like set
bool [:{TC3},{RR}:] is set
(TC2 .--> QQ) +* (TC3 .--> RR) is Relation-like Function-like set
TC2 .--> c100 is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> c100 is Relation-like {TC2} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{c100}:]
{c100} is non empty set
[:{TC2},{c100}:] is Relation-like set
bool [:{TC2},{c100}:] is set
((TC2,TC3) --> (QQ,RR)) +* (TC2 .--> c100) is Relation-like Function-like set
(TC2,TC3) --> (c100,RR) is Relation-like Function-like set
(TC2 .--> c100) +* (TC3 .--> RR) is Relation-like Function-like set
dom (TC2,TC3,TC2,QQ,RR,c100) is set
{TC2,TC3,TC2} is set
{TC2,TC2,TC3} is set
{TC2,TC3} is non empty set
dom ((TC2,TC3) --> (c100,RR)) is set
c100a is set
(TC2,TC3,TC2,QQ,RR,c100) . c100a is set
((TC2,TC3) --> (c100,RR)) . c100a is set
(TC3,TC2) --> (RR,c100) is Relation-like Function-like set
(TC3 .--> RR) +* (TC2 .--> c100) is Relation-like Function-like set
(TC2 .--> QQ) +* ((TC3,TC2) --> (RR,c100)) is Relation-like Function-like set
((TC2 .--> QQ) +* ((TC3,TC2) --> (RR,c100))) . c100a is set
TC2 is set
TC3 is set
QQ is set
RR is set
c100 is set
(TC2,TC3,TC3,QQ,RR,c100) is Relation-like Function-like set
(TC2,TC3) --> (QQ,RR) is Relation-like Function-like set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
{QQ} is non empty set
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
TC3 .--> RR is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> RR is Relation-like {TC3} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{RR}:]
{RR} is non empty set
[:{TC3},{RR}:] is Relation-like set
bool [:{TC3},{RR}:] is set
(TC2 .--> QQ) +* (TC3 .--> RR) is Relation-like Function-like set
TC3 .--> c100 is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} --> c100 is Relation-like {TC3} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{c100}:]
{c100} is non empty set
[:{TC3},{c100}:] is Relation-like set
bool [:{TC3},{c100}:] is set
((TC2,TC3) --> (QQ,RR)) +* (TC3 .--> c100) is Relation-like Function-like set
(TC2,TC3) --> (QQ,c100) is Relation-like Function-like set
(TC2 .--> QQ) +* (TC3 .--> c100) is Relation-like Function-like set
(TC3,TC3) --> (RR,c100) is Relation-like Function-like set
(TC3 .--> RR) +* (TC3 .--> c100) is Relation-like {TC3} -defined Function-like set
(TC2 .--> QQ) +* ((TC3,TC3) --> (RR,c100)) is Relation-like Function-like set
TC2 is set
TC3 is set
c100a is complex set
QQ is set
RR is set
c100 is set
(TC2,TC3,c100a,QQ,RR,c100) is Relation-like Function-like set
(TC2,TC3) --> (QQ,RR) is Relation-like Function-like set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
{QQ} is non empty set
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
TC3 .--> RR is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> RR is Relation-like {TC3} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{RR}:]
{RR} is non empty set
[:{TC3},{RR}:] is Relation-like set
bool [:{TC3},{RR}:] is set
(TC2 .--> QQ) +* (TC3 .--> RR) is Relation-like Function-like set
c100a .--> c100 is Relation-like {c100a} -defined Function-like one-to-one constant trivial set
{c100a} is non empty V195() set
{c100a} --> c100 is Relation-like {c100a} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{c100a},{c100}:]
{c100} is non empty set
[:{c100a},{c100}:] is Relation-like set
bool [:{c100a},{c100}:] is set
((TC2,TC3) --> (QQ,RR)) +* (c100a .--> c100) is Relation-like Function-like set
(TC2,TC3,c100a,QQ,RR,c100) . TC2 is set
TC2 is set
TC3 is set
QQ is set
RR is set
c100 is set
c100a is complex set
(TC2,TC3,c100a,QQ,RR,c100) is Relation-like Function-like set
(TC2,TC3) --> (QQ,RR) is Relation-like Function-like set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
{QQ} is non empty set
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
TC3 .--> RR is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> RR is Relation-like {TC3} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{RR}:]
{RR} is non empty set
[:{TC3},{RR}:] is Relation-like set
bool [:{TC3},{RR}:] is set
(TC2 .--> QQ) +* (TC3 .--> RR) is Relation-like Function-like set
c100a .--> c100 is Relation-like {c100a} -defined Function-like one-to-one constant trivial set
{c100a} is non empty V195() set
{c100a} --> c100 is Relation-like {c100a} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{c100a},{c100}:]
{c100} is non empty set
[:{c100a},{c100}:] is Relation-like set
bool [:{c100a},{c100}:] is set
((TC2,TC3) --> (QQ,RR)) +* (c100a .--> c100) is Relation-like Function-like set
(TC2,TC3,c100a,QQ,RR,c100) . TC2 is set
(TC2,TC3,c100a,QQ,RR,c100) . TC3 is set
(TC2,TC3,c100a,QQ,RR,c100) . c100a is set
dom (c100a .--> c100) is trivial V195() Element of bool {c100a}
bool {c100a} is V74() set
((TC2,TC3) --> (QQ,RR)) . TC2 is set
((TC2,TC3) --> (QQ,RR)) . TC3 is set
(c100a .--> c100) . c100a is set
TC2 is set
TC3 is set
QQ is set
RR is set
c100 is set
c100a is complex set
{TC2,TC3,c100a} is set
(TC2,TC3,c100a,QQ,RR,c100) is Relation-like Function-like set
(TC2,TC3) --> (QQ,RR) is Relation-like Function-like set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
{QQ} is non empty set
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
TC3 .--> RR is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> RR is Relation-like {TC3} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{RR}:]
{RR} is non empty set
[:{TC3},{RR}:] is Relation-like set
bool [:{TC3},{RR}:] is set
(TC2 .--> QQ) +* (TC3 .--> RR) is Relation-like Function-like set
c100a .--> c100 is Relation-like {c100a} -defined Function-like one-to-one constant trivial set
{c100a} is non empty V195() set
{c100a} --> c100 is Relation-like {c100a} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{c100a},{c100}:]
{c100} is non empty set
[:{c100a},{c100}:] is Relation-like set
bool [:{c100a},{c100}:] is set
((TC2,TC3) --> (QQ,RR)) +* (c100a .--> c100) is Relation-like Function-like set
mc100 is Relation-like Function-like set
dom mc100 is set
mc100 . TC2 is set
mc100 . TC3 is set
mc100 . c100a is set
dom (TC2,TC3,c100a,QQ,RR,c100) is set
SM is set
mc100 . SM is set
(TC2,TC3,c100a,QQ,RR,c100) . SM is set
(TC2,c100a) --> (RR,c100) is Relation-like Function-like set
TC2 .--> RR is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> RR is Relation-like {TC2} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{RR}:]
[:{TC2},{RR}:] is Relation-like set
bool [:{TC2},{RR}:] is set
(TC2 .--> RR) +* (c100a .--> c100) is Relation-like Function-like set
(TC2,TC3) --> (c100,RR) is Relation-like Function-like set
TC2 .--> c100 is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> c100 is Relation-like {TC2} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{c100}:]
[:{TC2},{c100}:] is Relation-like set
bool [:{TC2},{c100}:] is set
(TC2 .--> c100) +* (TC3 .--> RR) is Relation-like Function-like set
(TC2,TC3) --> (QQ,c100) is Relation-like Function-like set
TC3 .--> c100 is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} --> c100 is Relation-like {TC3} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{c100}:]
[:{TC3},{c100}:] is Relation-like set
bool [:{TC3},{c100}:] is set
(TC2 .--> QQ) +* (TC3 .--> c100) is Relation-like Function-like set
TC2 is set
TC3 is set
QQ is complex set
<*TC2,TC3,QQ*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like set
<*TC2*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like set
[1,TC2] is set
{1,TC2} is non empty set
{{1,TC2},{1}} is non empty set
{[1,TC2]} is Relation-like Function-like non empty set
<*TC3*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like set
[1,TC3] is set
{1,TC3} is non empty set
{{1,TC3},{1}} is non empty set
{[1,TC3]} is Relation-like Function-like non empty set
<*TC2*> ^ <*TC3*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like set
<*QQ*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding set
[1,QQ] is set
{1,QQ} is non empty V195() set
{{1,QQ},{1}} is non empty set
{[1,QQ]} is Relation-like Function-like non empty set
(<*TC2*> ^ <*TC3*>) ^ <*QQ*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like set
(1,2,3,TC2,TC3,QQ) is Relation-like Function-like set
(1,2) --> (TC2,TC3) is Relation-like REAL -defined Function-like set
1 .--> TC2 is Relation-like REAL -defined {1} -defined Function-like one-to-one constant trivial set
{1} --> TC2 is Relation-like {1} -defined {TC2} -valued Function-like constant non empty total quasi_total Element of bool [:{1},{TC2}:]
{TC2} is non empty set
[:{1},{TC2}:] is Relation-like set
bool [:{1},{TC2}:] is set
2 .--> TC3 is Relation-like REAL -defined {2} -defined Function-like one-to-one constant trivial set
{2} is non empty V195() V196() V197() V198() V199() V200() set
{2} --> TC3 is Relation-like {2} -defined {TC3} -valued Function-like constant non empty total quasi_total Element of bool [:{2},{TC3}:]
{TC3} is non empty set
[:{2},{TC3}:] is Relation-like set
bool [:{2},{TC3}:] is set
(1 .--> TC2) +* (2 .--> TC3) is Relation-like REAL -defined Function-like set
3 .--> QQ is Relation-like REAL -defined {3} -defined Function-like one-to-one constant trivial complex-yielding set
{3} is non empty V195() V196() V197() V198() V199() V200() set
{3} --> QQ is Relation-like {3} -defined {QQ} -valued Function-like constant non empty total quasi_total complex-yielding Element of bool [:{3},{QQ}:]
{QQ} is non empty V195() set
[:{3},{QQ}:] is Relation-like complex-yielding set
bool [:{3},{QQ}:] is set
((1,2) --> (TC2,TC3)) +* (3 .--> QQ) is Relation-like REAL -defined Function-like set
dom <*TC2,TC3,QQ*> is non empty V195() V196() V197() V198() V199() V200() Element of bool NAT
len <*TC2,TC3,QQ*> is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT
Seg (len <*TC2,TC3,QQ*>) is V32() V39( len <*TC2,TC3,QQ*>) V195() V196() V197() V198() V199() V200() Element of bool NAT
{ b₁ where b₁ is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT : ( 1 <= b₁ & b₁ <= len <*TC2,TC3,QQ*> ) } is set
dom (1,2,3,TC2,TC3,QQ) is set
c100a is set
(1,2,3,TC2,TC3,QQ) . c100a is set
<*TC2,TC3,QQ*> . c100a is set
(1,2,3,TC2,TC3,QQ) . c100a is set
<*TC2,TC3,QQ*> . c100a is set
(1,2,3,TC2,TC3,QQ) . c100a is set
<*TC2,TC3,QQ*> . c100a is set
TC2 is set
TC3 is set
QQ is set
{QQ} is non empty set
RR is set
{RR} is non empty set
c100 is set
{c100} is non empty set
c100a is complex set
(TC2,TC3,c100a,{QQ},{RR},{c100}) is Relation-like Function-like set
(TC2,TC3) --> ({QQ},{RR}) is Relation-like Function-like set
TC2 .--> {QQ} is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> {QQ} is Relation-like non-empty {TC2} -defined {{QQ}} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{{QQ}}:]
{{QQ}} is non empty set
[:{TC2},{{QQ}}:] is Relation-like set
bool [:{TC2},{{QQ}}:] is set
TC3 .--> {RR} is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> {RR} is Relation-like non-empty {TC3} -defined {{RR}} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{{RR}}:]
{{RR}} is non empty set
[:{TC3},{{RR}}:] is Relation-like set
bool [:{TC3},{{RR}}:] is set
(TC2 .--> {QQ}) +* (TC3 .--> {RR}) is Relation-like Function-like set
c100a .--> {c100} is Relation-like {c100a} -defined Function-like one-to-one constant trivial set
{c100a} is non empty V195() set
{c100a} --> {c100} is Relation-like non-empty {c100a} -defined {{c100}} -valued Function-like constant non empty total quasi_total Element of bool [:{c100a},{{c100}}:]
{{c100}} is non empty set
[:{c100a},{{c100}}:] is Relation-like set
bool [:{c100a},{{c100}}:] is set
((TC2,TC3) --> ({QQ},{RR})) +* (c100a .--> {c100}) is Relation-like Function-like set
product (TC2,TC3,c100a,{QQ},{RR},{c100}) is set
(TC2,TC3,c100a,QQ,RR,c100) is Relation-like Function-like set
(TC2,TC3) --> (QQ,RR) is Relation-like Function-like set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
TC3 .--> RR is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} --> RR is Relation-like {TC3} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{RR}:]
[:{TC3},{RR}:] is Relation-like set
bool [:{TC3},{RR}:] is set
(TC2 .--> QQ) +* (TC3 .--> RR) is Relation-like Function-like set
c100a .--> c100 is Relation-like {c100a} -defined Function-like one-to-one constant trivial set
{c100a} --> c100 is Relation-like {c100a} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{c100a},{c100}:]
[:{c100a},{c100}:] is Relation-like set
bool [:{c100a},{c100}:] is set
((TC2,TC3) --> (QQ,RR)) +* (c100a .--> c100) is Relation-like Function-like set
{(TC2,TC3,c100a,QQ,RR,c100)} is functional non empty set
dom (TC2,TC3,c100a,{QQ},{RR},{c100}) is set
{TC2,TC3,c100a} is set
SM is set
r is Relation-like Function-like set
dom r is set
p is set
r . p is set
(TC2,TC3,c100a,{QQ},{RR},{c100}) . p is set
r . TC2 is set
(TC2,TC3,c100a,{QQ},{RR},{c100}) . TC2 is set
r . TC3 is set
(TC2,TC3,c100a,{QQ},{RR},{c100}) . TC3 is set
r . c100a is set
(TC2,TC3,c100a,{QQ},{RR},{c100}) . c100a is set
r is Relation-like Function-like set
dom r is set
r . TC2 is set
(TC2,TC3,c100a,{QQ},{RR},{c100}) . TC2 is set
r . TC3 is set
(TC2,TC3,c100a,{QQ},{RR},{c100}) . TC3 is set
r . c100a is set
(TC2,TC3,c100a,{QQ},{RR},{c100}) . c100a is set
TC2 is set
TC3 is set
QQ is complex set
RR is set
c100 is set
c100a is set
mc100 is set
CM is set
SM is set
(TC2,TC3,QQ,RR,c100a,CM) is Relation-like Function-like set
(TC2,TC3) --> (RR,c100a) is Relation-like Function-like set
TC2 .--> RR is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> RR is Relation-like {TC2} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{RR}:]
{RR} is non empty set
[:{TC2},{RR}:] is Relation-like set
bool [:{TC2},{RR}:] is set
TC3 .--> c100a is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> c100a is Relation-like {TC3} -defined {c100a} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{c100a}:]
{c100a} is non empty set
[:{TC3},{c100a}:] is Relation-like set
bool [:{TC3},{c100a}:] is set
(TC2 .--> RR) +* (TC3 .--> c100a) is Relation-like Function-like set
QQ .--> CM is Relation-like {QQ} -defined Function-like one-to-one constant trivial set
{QQ} is non empty V195() set
{QQ} --> CM is Relation-like {QQ} -defined {CM} -valued Function-like constant non empty total quasi_total Element of bool [:{QQ},{CM}:]
{CM} is non empty set
[:{QQ},{CM}:] is Relation-like set
bool [:{QQ},{CM}:] is set
((TC2,TC3) --> (RR,c100a)) +* (QQ .--> CM) is Relation-like Function-like set
product (TC2,TC3,QQ,RR,c100a,CM) is set
(TC2,TC3,QQ,c100,mc100,SM) is Relation-like Function-like set
(TC2,TC3) --> (c100,mc100) is Relation-like Function-like set
TC2 .--> c100 is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> c100 is Relation-like {TC2} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{c100}:]
{c100} is non empty set
[:{TC2},{c100}:] is Relation-like set
bool [:{TC2},{c100}:] is set
TC3 .--> mc100 is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} --> mc100 is Relation-like {TC3} -defined {mc100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{mc100}:]
{mc100} is non empty set
[:{TC3},{mc100}:] is Relation-like set
bool [:{TC3},{mc100}:] is set
(TC2 .--> c100) +* (TC3 .--> mc100) is Relation-like Function-like set
QQ .--> SM is Relation-like {QQ} -defined Function-like one-to-one constant trivial set
{QQ} --> SM is Relation-like {QQ} -defined {SM} -valued Function-like constant non empty total quasi_total Element of bool [:{QQ},{SM}:]
{SM} is non empty set
[:{QQ},{SM}:] is Relation-like set
bool [:{QQ},{SM}:] is set
((TC2,TC3) --> (c100,mc100)) +* (QQ .--> SM) is Relation-like Function-like set
product (TC2,TC3,QQ,c100,mc100,SM) is set
dom (TC2,TC3,QQ,RR,c100a,CM) is set
{TC2,TC3,QQ} is set
dom (TC2,TC3,QQ,c100,mc100,SM) is set
(TC2,TC3) --> (CM,c100a) is Relation-like Function-like set
TC2 .--> CM is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> CM is Relation-like {TC2} -defined {CM} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{CM}:]
[:{TC2},{CM}:] is Relation-like set
bool [:{TC2},{CM}:] is set
(TC2 .--> CM) +* (TC3 .--> c100a) is Relation-like Function-like set
(TC2,TC3) --> (SM,mc100) is Relation-like Function-like set
TC2 .--> SM is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> SM is Relation-like {TC2} -defined {SM} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{SM}:]
[:{TC2},{SM}:] is Relation-like set
bool [:{TC2},{SM}:] is set
(TC2 .--> SM) +* (TC3 .--> mc100) is Relation-like Function-like set
(TC2,TC3) --> (RR,CM) is Relation-like Function-like set
TC3 .--> CM is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} --> CM is Relation-like {TC3} -defined {CM} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{CM}:]
[:{TC3},{CM}:] is Relation-like set
bool [:{TC3},{CM}:] is set
(TC2 .--> RR) +* (TC3 .--> CM) is Relation-like Function-like set
(TC2,TC3) --> (c100,SM) is Relation-like Function-like set
TC3 .--> SM is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} --> SM is Relation-like {TC3} -defined {SM} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{SM}:]
[:{TC3},{SM}:] is Relation-like set
bool [:{TC3},{SM}:] is set
(TC2 .--> c100) +* (TC3 .--> SM) is Relation-like Function-like set
(TC2,QQ) --> (c100a,CM) is Relation-like Function-like set
TC2 .--> c100a is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> c100a is Relation-like {TC2} -defined {c100a} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{c100a}:]
[:{TC2},{c100a}:] is Relation-like set
bool [:{TC2},{c100a}:] is set
(TC2 .--> c100a) +* (QQ .--> CM) is Relation-like Function-like set
(TC2,QQ) --> (mc100,SM) is Relation-like Function-like set
TC2 .--> mc100 is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> mc100 is Relation-like {TC2} -defined {mc100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{mc100}:]
[:{TC2},{mc100}:] is Relation-like set
bool [:{TC2},{mc100}:] is set
(TC2 .--> mc100) +* (QQ .--> SM) is Relation-like Function-like set
f is set
(TC2,TC3,QQ,RR,c100a,CM) . f is set
(TC2,TC3,QQ,c100,mc100,SM) . f is set
(TC2,TC3,QQ,RR,c100a,CM) . TC2 is set
(TC2,TC3,QQ,RR,c100a,CM) . TC3 is set
(TC2,TC3,QQ,RR,c100a,CM) . QQ is set
(TC2,TC3,QQ,c100,mc100,SM) . TC2 is set
(TC2,TC3,QQ,c100,mc100,SM) . TC3 is set
(TC2,TC3,QQ,c100,mc100,SM) . QQ is set
TC2 is set
TC3 is set
QQ is set
RR is set
c100 is set
c100a is set
mc100 is set
CM is set
SM is complex set
(TC2,TC3,SM,QQ,c100,mc100) is Relation-like Function-like set
(TC2,TC3) --> (QQ,c100) is Relation-like Function-like set
TC2 .--> QQ is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} is non empty set
{TC2} --> QQ is Relation-like {TC2} -defined {QQ} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{QQ}:]
{QQ} is non empty set
[:{TC2},{QQ}:] is Relation-like set
bool [:{TC2},{QQ}:] is set
TC3 .--> c100 is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} is non empty set
{TC3} --> c100 is Relation-like {TC3} -defined {c100} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{c100}:]
{c100} is non empty set
[:{TC3},{c100}:] is Relation-like set
bool [:{TC3},{c100}:] is set
(TC2 .--> QQ) +* (TC3 .--> c100) is Relation-like Function-like set
SM .--> mc100 is Relation-like {SM} -defined Function-like one-to-one constant trivial set
{SM} is non empty V195() set
{SM} --> mc100 is Relation-like {SM} -defined {mc100} -valued Function-like constant non empty total quasi_total Element of bool [:{SM},{mc100}:]
{mc100} is non empty set
[:{SM},{mc100}:] is Relation-like set
bool [:{SM},{mc100}:] is set
((TC2,TC3) --> (QQ,c100)) +* (SM .--> mc100) is Relation-like Function-like set
(TC2,TC3,SM,RR,c100a,CM) is Relation-like Function-like set
(TC2,TC3) --> (RR,c100a) is Relation-like Function-like set
TC2 .--> RR is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> RR is Relation-like {TC2} -defined {RR} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{RR}:]
{RR} is non empty set
[:{TC2},{RR}:] is Relation-like set
bool [:{TC2},{RR}:] is set
TC3 .--> c100a is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} --> c100a is Relation-like {TC3} -defined {c100a} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{c100a}:]
{c100a} is non empty set
[:{TC3},{c100a}:] is Relation-like set
bool [:{TC3},{c100a}:] is set
(TC2 .--> RR) +* (TC3 .--> c100a) is Relation-like Function-like set
SM .--> CM is Relation-like {SM} -defined Function-like one-to-one constant trivial set
{SM} --> CM is Relation-like {SM} -defined {CM} -valued Function-like constant non empty total quasi_total Element of bool [:{SM},{CM}:]
{CM} is non empty set
[:{SM},{CM}:] is Relation-like set
bool [:{SM},{CM}:] is set
((TC2,TC3) --> (RR,c100a)) +* (SM .--> CM) is Relation-like Function-like set
product (TC2,TC3,SM,RR,c100a,CM) is set
(TC2,TC3,SM,{QQ},{c100},{mc100}) is Relation-like Function-like set
(TC2,TC3) --> ({QQ},{c100}) is Relation-like Function-like set
TC2 .--> {QQ} is Relation-like {TC2} -defined Function-like one-to-one constant trivial set
{TC2} --> {QQ} is Relation-like non-empty {TC2} -defined {{QQ}} -valued Function-like constant non empty total quasi_total Element of bool [:{TC2},{{QQ}}:]
{{QQ}} is non empty set
[:{TC2},{{QQ}}:] is Relation-like set
bool [:{TC2},{{QQ}}:] is set
TC3 .--> {c100} is Relation-like {TC3} -defined Function-like one-to-one constant trivial set
{TC3} --> {c100} is Relation-like non-empty {TC3} -defined {{c100}} -valued Function-like constant non empty total quasi_total Element of bool [:{TC3},{{c100}}:]
{{c100}} is non empty set
[:{TC3},{{c100}}:] is Relation-like set
bool [:{TC3},{{c100}}:] is set
(TC2 .--> {QQ}) +* (TC3 .--> {c100}) is Relation-like Function-like set
SM .--> {mc100} is Relation-like {SM} -defined Function-like one-to-one constant trivial set
{SM} --> {mc100} is Relation-like non-empty {SM} -defined {{mc100}} -valued Function-like constant non empty total quasi_total Element of bool [:{SM},{{mc100}}:]
{{mc100}} is non empty set
[:{SM},{{mc100}}:] is Relation-like set
bool [:{SM},{{mc100}}:] is set
((TC2,TC3) --> ({QQ},{c100})) +* (SM .--> {mc100}) is Relation-like Function-like set
product (TC2,TC3,SM,{QQ},{c100},{mc100}) is set
{(TC2,TC3,SM,QQ,c100,mc100)} is functional non empty set
dom {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V32() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued Element of bool NAT
TC2 is Relation-like Function-like complex-yielding set
- TC2 is Relation-like Function-like complex-yielding set
- 1 is non empty complex real integer ext-real non positive negative set
(- 1) * TC2 is Relation-like Function-like complex-yielding set
TC3 is Relation-like Function-like complex-yielding set
{} . TC2 is Relation-like non-empty empty-yielding NAT -defined RAT -valued INT -valued Function-like one-to-one constant functional empty complex ordinal natural real integer non irrational V32() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued set
{} . (- TC2) is Relation-like non-empty empty-yielding NAT -defined RAT -valued INT -valued Function-like one-to-one constant functional empty complex ordinal natural real integer non irrational V32() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued set
- TC3 is Relation-like Function-like complex-yielding set
(- 1) * TC3 is Relation-like Function-like complex-yielding set
Tunit_circle 3 is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 3
Tcircle ((0. (TOP-REAL 3)),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 3
Sphere ((0. (TOP-REAL 3)),1) is functional non empty closed V214( TOP-REAL 3) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 3)
bool the carrier of (TOP-REAL 3) is V74() set
(TOP-REAL 3) | (Sphere ((0. (TOP-REAL 3)),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 3
the carrier of (Tunit_circle 3) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
Sphere (|[0,0,0]|,1) is functional non empty closed V214( TOP-REAL 3) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 3)
QQ is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
sqr QQ is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
mlt (QQ,QQ) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (QQ,QQ) is Relation-like Function-like set
K254(QQ,QQ) is Relation-like Function-like set
K254(QQ,QQ) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
QQ (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
QQ `1 is complex real ext-real Element of REAL
(QQ `1) ^2 is complex real ext-real non negative Element of REAL
(QQ `1) * (QQ `1) is complex real ext-real non negative set
QQ `2 is complex real ext-real Element of REAL
(QQ `2) ^2 is complex real ext-real non negative Element of REAL
(QQ `2) * (QQ `2) is complex real ext-real non negative set
QQ `3 is complex real ext-real Element of REAL
(QQ `3) ^2 is complex real ext-real non negative Element of REAL
(QQ `3) * (QQ `3) is complex real ext-real non negative set
<*((QQ `1) ^2),((QQ `2) ^2),((QQ `3) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((QQ `1) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((QQ `1) ^2)] is set
{1,((QQ `1) ^2)} is non empty V195() V196() V197() set
{{1,((QQ `1) ^2)},{1}} is non empty set
{[1,((QQ `1) ^2)]} is Relation-like Function-like non empty set
<*((QQ `2) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((QQ `2) ^2)] is set
{1,((QQ `2) ^2)} is non empty V195() V196() V197() set
{{1,((QQ `2) ^2)},{1}} is non empty set
{[1,((QQ `2) ^2)]} is Relation-like Function-like non empty set
<*((QQ `1) ^2)*> ^ <*((QQ `2) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((QQ `3) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((QQ `3) ^2)] is set
{1,((QQ `3) ^2)} is non empty V195() V196() V197() set
{{1,((QQ `3) ^2)},{1}} is non empty set
{[1,((QQ `3) ^2)]} is Relation-like Function-like non empty set
(<*((QQ `1) ^2)*> ^ <*((QQ `2) ^2)*>) ^ <*((QQ `3) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[(QQ `1),(QQ `2),(QQ `3)]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(QQ `1)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(QQ `1)] is set
{1,(QQ `1)} is non empty V195() V196() V197() set
{{1,(QQ `1)},{1}} is non empty set
{[1,(QQ `1)]} is Relation-like Function-like non empty set
<*(QQ `2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(QQ `2)] is set
{1,(QQ `2)} is non empty V195() V196() V197() set
{{1,(QQ `2)},{1}} is non empty set
{[1,(QQ `2)]} is Relation-like Function-like non empty set
<*(QQ `1)*> ^ <*(QQ `2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(QQ `3)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(QQ `3)] is set
{1,(QQ `3)} is non empty V195() V196() V197() set
{{1,(QQ `3)},{1}} is non empty set
{[1,(QQ `3)]} is Relation-like Function-like non empty set
(<*(QQ `1)*> ^ <*(QQ `2)*>) ^ <*(QQ `3)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
QQ is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
sqr QQ is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
mlt (QQ,QQ) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (QQ,QQ) is Relation-like Function-like set
K254(QQ,QQ) is Relation-like Function-like set
K254(QQ,QQ) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
QQ (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr QQ) is complex real ext-real set
QQ `1 is complex real ext-real Element of REAL
(QQ `1) ^2 is complex real ext-real non negative Element of REAL
(QQ `1) * (QQ `1) is complex real ext-real non negative set
QQ `2 is complex real ext-real Element of REAL
(QQ `2) ^2 is complex real ext-real non negative Element of REAL
(QQ `2) * (QQ `2) is complex real ext-real non negative set
((QQ `1) ^2) + ((QQ `2) ^2) is complex real ext-real non negative Element of REAL
QQ `3 is complex real ext-real Element of REAL
(QQ `3) ^2 is complex real ext-real non negative Element of REAL
(QQ `3) * (QQ `3) is complex real ext-real non negative set
(((QQ `1) ^2) + ((QQ `2) ^2)) + ((QQ `3) ^2) is complex real ext-real non negative Element of REAL
|[(QQ `1),(QQ `2),(QQ `3)]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(QQ `1)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(QQ `1)] is set
{1,(QQ `1)} is non empty V195() V196() V197() set
{{1,(QQ `1)},{1}} is non empty set
{[1,(QQ `1)]} is Relation-like Function-like non empty set
<*(QQ `2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(QQ `2)] is set
{1,(QQ `2)} is non empty V195() V196() V197() set
{{1,(QQ `2)},{1}} is non empty set
{[1,(QQ `2)]} is Relation-like Function-like non empty set
<*(QQ `1)*> ^ <*(QQ `2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(QQ `3)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(QQ `3)] is set
{1,(QQ `3)} is non empty V195() V196() V197() set
{{1,(QQ `3)},{1}} is non empty set
{[1,(QQ `3)]} is Relation-like Function-like non empty set
(<*(QQ `1)*> ^ <*(QQ `2)*>) ^ <*(QQ `3)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
QQ is V195() V196() V197() Element of bool REAL
R^1 QQ is V195() V196() V197() Element of bool the carrier of K708()
R^1 | (R^1 QQ) is strict TopSpace-like V231() SubSpace of R^1
the carrier of (R^1 | (R^1 QQ)) is V195() V196() V197() set
c100 is complex real ext-real set
].-infty,c100.[ is non empty V195() V196() V197() Element of bool REAL
RAT /\ ].-infty,c100.[ is V195() V196() V197() V198() Element of bool REAL
c100a is V195() V196() V197() Element of bool the carrier of R^1
R^1 | c100a is strict TopSpace-like V231() SubSpace of R^1
the carrier of (R^1 | c100a) is V195() V196() V197() set
bool the carrier of (R^1 | c100a) is set
bool the carrier of (R^1 | (R^1 QQ)) is set
R^1 ].-infty,c100.[ is non empty V195() V196() V197() Element of bool the carrier of K708()
CM is V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 QQ))
(R^1 ].-infty,c100.[) /\ the carrier of (R^1 | (R^1 QQ)) is V195() V196() V197() Element of bool the carrier of K708()
c100 is complex real ext-real set
].c100,+infty.[ is non empty V195() V196() V197() Element of bool REAL
RAT /\ ].c100,+infty.[ is V195() V196() V197() V198() Element of bool REAL
c100a is V195() V196() V197() Element of bool the carrier of R^1
R^1 | c100a is strict TopSpace-like V231() SubSpace of R^1
the carrier of (R^1 | c100a) is V195() V196() V197() set
bool the carrier of (R^1 | c100a) is set
bool the carrier of (R^1 | (R^1 QQ)) is set
R^1 ].c100,+infty.[ is non empty V195() V196() V197() Element of bool the carrier of K708()
CM is V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 QQ))
(R^1 ].c100,+infty.[) /\ the carrier of (R^1 | (R^1 QQ)) is V195() V196() V197() Element of bool the carrier of K708()
c100 is non empty TopSpace-like connected TopStruct
the carrier of c100 is non empty set
[: the carrier of c100, the carrier of (R^1 | (R^1 QQ)):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of c100, the carrier of (R^1 | (R^1 QQ)):] is set
mc100 is set
c100a is Relation-like the carrier of c100 -defined the carrier of (R^1 | (R^1 QQ)) -valued Function-like quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of c100, the carrier of (R^1 | (R^1 QQ)):]
dom c100a is Element of bool the carrier of c100
bool the carrier of c100 is V74() set
CM is set
c100a . CM is complex real ext-real set
c100a . mc100 is complex real ext-real set
SM is non empty complex real irrational ext-real set
Image c100a is TopSpace-like V231() SubSpace of R^1 | (R^1 QQ)
rng c100a is V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 QQ))
bool the carrier of (R^1 | (R^1 QQ)) is set
[#] (Image c100a) is non proper open closed dense V195() V196() V197() Element of bool the carrier of (Image c100a)
the carrier of (Image c100a) is V195() V196() V197() set
bool the carrier of (Image c100a) is set
{} (Image c100a) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V32() open closed boundary nowhere_dense FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued Element of bool the carrier of (Image c100a)
{ b₁ where b₁ is complex real non irrational ext-real Element of RAT : ( b₁ in rng c100a & not SM <= b₁ ) } is set
{ b₁ where b₁ is complex real non irrational ext-real Element of RAT : ( b₁ in rng c100a & not b₁ <= SM ) } is set
e is set
f1 is complex real non irrational ext-real Element of RAT
f1 is set
h is complex real non irrational ext-real Element of RAT
e is V195() V196() V197() Element of bool the carrier of (Image c100a)
f1 is V195() V196() V197() Element of bool the carrier of (Image c100a)
e \/ f1 is V195() V196() V197() Element of bool the carrier of (Image c100a)
h is set
x is complex real non irrational ext-real Element of RAT
y is complex real non irrational ext-real Element of RAT
].-infty,SM.[ is non empty V195() V196() V197() Element of bool REAL
RAT /\ ].-infty,SM.[ is V195() V196() V197() V198() Element of bool REAL
].SM,+infty.[ is non empty V195() V196() V197() Element of bool REAL
RAT /\ ].SM,+infty.[ is V195() V196() V197() V198() Element of bool REAL
h is open V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 QQ))
h /\ the carrier of (Image c100a) is V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 QQ))
y is set
hx is complex real non irrational ext-real Element of RAT
y is set
hx is complex real non irrational ext-real Element of RAT
x is open V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 QQ))
x /\ the carrier of (Image c100a) is V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 QQ))
y is set
hx is complex real non irrational ext-real Element of RAT
y is set
hx is complex real non irrational ext-real Element of RAT
y is complex real ext-real Element of the carrier of (Image c100a)
c100 is non empty TopSpace-like connected TopStruct
the carrier of c100 is non empty set
c100a is non empty TopSpace-like TopStruct
the carrier of c100a is non empty set
[: the carrier of c100, the carrier of c100a:] is Relation-like set
bool [: the carrier of c100, the carrier of c100a:] is set
mc100 is Relation-like the carrier of c100 -defined the carrier of c100a -valued Function-like non empty total quasi_total continuous Element of bool [: the carrier of c100, the carrier of c100a:]
Image mc100 is non empty TopSpace-like SubSpace of c100a
corestr mc100 is Relation-like the carrier of c100 -defined the carrier of (Image mc100) -valued Function-like non empty total quasi_total onto Element of bool [: the carrier of c100, the carrier of (Image mc100):]
the carrier of (Image mc100) is non empty set
[: the carrier of c100, the carrier of (Image mc100):] is Relation-like set
bool [: the carrier of c100, the carrier of (Image mc100):] is set
dom (corestr mc100) is non empty Element of bool the carrier of c100
bool the carrier of c100 is V74() set
[#] c100 is non empty non proper open closed dense non boundary Element of bool the carrier of c100
rng (corestr mc100) is non empty Element of bool the carrier of (Image mc100)
bool the carrier of (Image mc100) is V74() set
[#] (Image mc100) is non empty non proper open closed dense non boundary Element of bool the carrier of (Image mc100)
(corestr mc100) .: ([#] c100) is Element of bool the carrier of (Image mc100)
c100 is V195() V196() V197() Element of bool the carrier of R^1
R^1 | c100 is strict TopSpace-like V231() SubSpace of R^1
the carrier of (R^1 | c100) is V195() V196() V197() set
c100a is TopSpace-like connected TopStruct
the carrier of c100a is set
[: the carrier of c100a, the carrier of (R^1 | c100):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of c100a, the carrier of (R^1 | c100):] is set
mc100 is Relation-like the carrier of c100a -defined the carrier of (R^1 | c100) -valued Function-like quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of c100a, the carrier of (R^1 | c100):]
Image mc100 is TopSpace-like V231() SubSpace of R^1 | c100
SM is set
dom mc100 is set
r is set
mc100 . SM is complex real ext-real set
mc100 . r is complex real ext-real set
dom mc100 is Element of bool the carrier of c100a
bool the carrier of c100a is set
the carrier of (Image mc100) is V195() V196() V197() set
bool the carrier of (Image mc100) is set
{} (Image mc100) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V32() open closed boundary nowhere_dense FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued Element of bool the carrier of (Image mc100)
[#] (Image mc100) is non proper open closed dense V195() V196() V197() Element of bool the carrier of (Image mc100)
p is V195() V196() V197() Element of bool the carrier of (Image mc100)
f is V195() V196() V197() Element of bool the carrier of (Image mc100)
p \/ f is V195() V196() V197() Element of bool the carrier of (Image mc100)
the carrier of (Image mc100) is V195() V196() V197() set
bool the carrier of (Image mc100) is set
{} (Image mc100) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V32() open closed boundary nowhere_dense FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued Element of bool the carrier of (Image mc100)
[#] (Image mc100) is non proper open closed dense V195() V196() V197() Element of bool the carrier of (Image mc100)
p is V195() V196() V197() Element of bool the carrier of (Image mc100)
f is V195() V196() V197() Element of bool the carrier of (Image mc100)
p \/ f is V195() V196() V197() Element of bool the carrier of (Image mc100)
c100a is complex real ext-real set
mc100 is complex real ext-real set
Closed-Interval-TSpace (c100a,mc100) is non empty strict TopSpace-like V231() SubSpace of R^1
the carrier of (Closed-Interval-TSpace (c100a,mc100)) is non empty V195() V196() V197() set
[: the carrier of (Closed-Interval-TSpace (c100a,mc100)), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (Closed-Interval-TSpace (c100a,mc100)), the carrier of R^1:] is set
SM is Relation-like the carrier of (Closed-Interval-TSpace (c100a,mc100)) -defined the carrier of R^1 -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (Closed-Interval-TSpace (c100a,mc100)), the carrier of R^1:]
r is Relation-like REAL -defined REAL -valued Function-like complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
left_closed_halfline c100a is V195() V196() V197() closed Element of bool REAL
K699(-infty,c100a) is non empty set
[.c100a,mc100.] is V195() V196() V197() closed Element of bool REAL
right_closed_halfline mc100 is V195() V196() V197() closed Element of bool REAL
K698(mc100,+infty) is non empty set
R^1 (left_closed_halfline c100a) is closed V195() V196() V197() Element of bool the carrier of K708()
R^1 | (R^1 (left_closed_halfline c100a)) is strict TopSpace-like closed V231() SubSpace of R^1
f1 is complex real ext-real Element of the carrier of (Closed-Interval-TSpace (c100a,mc100))
SM . f1 is complex real ext-real Element of the carrier of R^1
(R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1) is Relation-like the carrier of (R^1 | (R^1 (left_closed_halfline c100a))) -defined the carrier of R^1 -valued Function-like total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (R^1 | (R^1 (left_closed_halfline c100a))), the carrier of R^1:]
the carrier of (R^1 | (R^1 (left_closed_halfline c100a))) is V195() V196() V197() set
[: the carrier of (R^1 | (R^1 (left_closed_halfline c100a))), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (R^1 | (R^1 (left_closed_halfline c100a))), the carrier of R^1:] is set
the carrier of (R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1) is Relation-like the carrier of (R^1 | (R^1 (left_closed_halfline c100a))) -defined the carrier of R^1 -valued Function-like constant total quasi_total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (R^1 | (R^1 (left_closed_halfline c100a))), the carrier of R^1:]
{(SM . f1)} is non empty V195() V196() V197() set
[: the carrier of (R^1 | (R^1 (left_closed_halfline c100a))),{(SM . f1)}:] is Relation-like complex-yielding ext-real-valued real-valued set
R^1 (right_closed_halfline mc100) is closed V195() V196() V197() Element of bool the carrier of K708()
R^1 | (R^1 (right_closed_halfline mc100)) is strict TopSpace-like closed V231() SubSpace of R^1
h is complex real ext-real Element of the carrier of (Closed-Interval-TSpace (c100a,mc100))
SM . h is complex real ext-real Element of the carrier of R^1
(R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h) is Relation-like the carrier of (R^1 | (R^1 (right_closed_halfline mc100))) -defined the carrier of R^1 -valued Function-like total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (R^1 | (R^1 (right_closed_halfline mc100))), the carrier of R^1:]
the carrier of (R^1 | (R^1 (right_closed_halfline mc100))) is V195() V196() V197() set
[: the carrier of (R^1 | (R^1 (right_closed_halfline mc100))), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (R^1 | (R^1 (right_closed_halfline mc100))), the carrier of R^1:] is set
the carrier of (R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h) is Relation-like the carrier of (R^1 | (R^1 (right_closed_halfline mc100))) -defined the carrier of R^1 -valued Function-like constant total quasi_total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (R^1 | (R^1 (right_closed_halfline mc100))), the carrier of R^1:]
{(SM . h)} is non empty V195() V196() V197() set
[: the carrier of (R^1 | (R^1 (right_closed_halfline mc100))),{(SM . h)}:] is Relation-like complex-yielding ext-real-valued real-valued set
R^1 [.c100a,mc100.] is closed V195() V196() V197() Element of bool the carrier of K708()
R^1 | (R^1 [.c100a,mc100.]) is strict TopSpace-like closed V231() SubSpace of R^1
the carrier of (R^1 | (R^1 [.c100a,mc100.])) is V195() V196() V197() set
[: the carrier of (R^1 | (R^1 [.c100a,mc100.])), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (R^1 | (R^1 [.c100a,mc100.])), the carrier of R^1:] is set
dom ((R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1)) is V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 (left_closed_halfline c100a)))
bool the carrier of (R^1 | (R^1 (left_closed_halfline c100a))) is set
hx is Relation-like the carrier of (R^1 | (R^1 [.c100a,mc100.])) -defined the carrier of R^1 -valued Function-like total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (R^1 | (R^1 [.c100a,mc100.])), the carrier of R^1:]
dom hx is V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 [.c100a,mc100.]))
bool the carrier of (R^1 | (R^1 [.c100a,mc100.])) is set
dom ((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h)) is V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 (right_closed_halfline mc100)))
bool the carrier of (R^1 | (R^1 (right_closed_halfline mc100))) is set
hy is set
dom ((R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1)) is set
dom hx is set
(dom ((R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1))) /\ (dom hx) is set
((R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1)) . hy is complex real ext-real set
hx . hy is complex real ext-real set
(dom ((R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1))) /\ (dom hx) is V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 [.c100a,mc100.]))
(left_closed_halfline c100a) /\ [.c100a,mc100.] is V195() V196() V197() Element of bool REAL
{c100a} is non empty V195() V196() V197() set
(R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.]) is closed V195() V196() V197() Element of bool the carrier of K708()
R^1 | ((R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.])) is strict TopSpace-like V231() SubSpace of R^1
the carrier of (R^1 | ((R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.]))) is V195() V196() V197() set
[: the carrier of (R^1 | ((R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.]))), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (R^1 | ((R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.]))), the carrier of R^1:] is set
((R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1)) +* hx is Relation-like the carrier of R^1 -valued Function-like complex-yielding ext-real-valued real-valued set
hy is Relation-like the carrier of (R^1 | ((R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.]))) -defined the carrier of R^1 -valued Function-like total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (R^1 | ((R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.]))), the carrier of R^1:]
hy +* ((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h)) is Relation-like the carrier of R^1 -valued Function-like complex-yielding ext-real-valued real-valued set
Ch is set
dom hy is set
dom ((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h)) is set
(dom hy) /\ (dom ((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h))) is set
hy . Ch is complex real ext-real set
((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h)) . Ch is complex real ext-real set
dom hy is V195() V196() V197() Element of bool the carrier of (R^1 | ((R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.])))
bool the carrier of (R^1 | ((R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.]))) is set
(dom hy) /\ (dom ((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h))) is V195() V196() V197() Element of bool the carrier of (R^1 | (R^1 (right_closed_halfline mc100)))
X is complex real ext-real Element of REAL
(dom ((R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1))) \/ (dom hx) is V195() V196() V197() set
hx . f1 is complex real ext-real set
((R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1)) . Ch is complex real ext-real set
hx . Ch is complex real ext-real set
(left_closed_halfline c100a) \/ [.c100a,mc100.] is V195() V196() V197() Element of bool REAL
R^1 ((left_closed_halfline c100a) \/ [.c100a,mc100.]) is V195() V196() V197() Element of bool the carrier of K708()
(R^1 ((left_closed_halfline c100a) \/ [.c100a,mc100.])) \/ (R^1 (right_closed_halfline mc100)) is V195() V196() V197() Element of bool the carrier of K708()
R^1 | ((R^1 ((left_closed_halfline c100a) \/ [.c100a,mc100.])) \/ (R^1 (right_closed_halfline mc100))) is strict TopSpace-like V231() SubSpace of R^1
the carrier of (R^1 | ((R^1 ((left_closed_halfline c100a) \/ [.c100a,mc100.])) \/ (R^1 (right_closed_halfline mc100)))) is V195() V196() V197() set
[: the carrier of (R^1 | ((R^1 ((left_closed_halfline c100a) \/ [.c100a,mc100.])) \/ (R^1 (right_closed_halfline mc100)))), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (R^1 | ((R^1 ((left_closed_halfline c100a) \/ [.c100a,mc100.])) \/ (R^1 (right_closed_halfline mc100)))), the carrier of R^1:] is set
((left_closed_halfline c100a) \/ [.c100a,mc100.]) \/ (right_closed_halfline mc100) is V195() V196() V197() Element of bool REAL
dom (hy +* ((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h))) is set
dom hy is V195() V196() V197() Element of bool the carrier of (R^1 | ((R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.])))
bool the carrier of (R^1 | ((R^1 (left_closed_halfline c100a)) \/ (R^1 [.c100a,mc100.]))) is set
(dom hy) \/ (dom ((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h))) is V195() V196() V197() set
(dom ((R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1))) \/ (dom hx) is V195() V196() V197() set
((dom ((R^1 | (R^1 (left_closed_halfline c100a))) --> (SM . f1))) \/ (dom hx)) \/ (dom ((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h))) is V195() V196() V197() set
Ch is set
X is complex real ext-real Element of REAL
[#] R^1 is non empty non proper open closed dense non boundary V195() V196() V197() Element of bool the carrier of R^1
rng (hy +* ((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h))) is V195() V196() V197() Element of bool the carrier of R^1
Ch is Relation-like REAL -defined REAL -valued Function-like complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
dom Ch is V195() V196() V197() Element of bool REAL
(dom Ch) /\ [.c100a,mc100.] is V195() V196() V197() Element of bool REAL
X is set
xx is complex real ext-real Element of REAL
Ch . X is complex real ext-real set
hy . X is complex real ext-real set
hx . X is complex real ext-real set
Ch . X is complex real ext-real set
((R^1 | (R^1 (right_closed_halfline mc100))) --> (SM . h)) . X is complex real ext-real set
hx . X is complex real ext-real set
Ch | [.c100a,mc100.] is Relation-like REAL -defined [.c100a,mc100.] -defined REAL -defined REAL -valued Function-like complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
c100 is complex real ext-real Element of the carrier of R^1
c100a is complex real ext-real set
c100 + c100a is complex real ext-real set
c100 is complex real ext-real Element of the carrier of R^1
c100a is complex real ext-real set
c100 - c100a is complex real ext-real set
- c100a is complex real ext-real set
c100 + (- c100a) is complex real ext-real set
c100 is set
[:c100, the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:c100, the carrier of R^1:] is set
c100a is Relation-like c100 -defined the carrier of R^1 -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:c100, the carrier of R^1:]
mc100 is complex real ext-real set
mc100 + c100a is Relation-like c100 -defined Function-like total complex-yielding ext-real-valued real-valued set
mc100 + c100a is Relation-like c100 -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:c100,REAL:]
[:c100,REAL:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:c100,REAL:] is set
c100 is set
[:c100, the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:c100, the carrier of R^1:] is set
c100a is Relation-like c100 -defined the carrier of R^1 -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:c100, the carrier of R^1:]
mc100 is complex real ext-real set
c100a - mc100 is Relation-like c100 -defined Function-like total complex-yielding ext-real-valued real-valued set
- mc100 is complex real ext-real set
(- mc100) + c100a is Relation-like c100 -defined Function-like total complex-yielding ext-real-valued real-valued set
c100a - mc100 is Relation-like c100 -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:c100,REAL:]
[:c100,REAL:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:c100,REAL:] is set
c100 is complex real ext-real Element of the carrier of R^1
c100a is complex real ext-real Element of the carrier of R^1
mc100 is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued Path of c100,c100a
CM is complex real ext-real set
CM + mc100 is Relation-like the carrier of I[01] -defined Function-like total complex-yielding ext-real-valued real-valued set
(c100,CM) is complex real ext-real Element of the carrier of R^1
(c100a,CM) is complex real ext-real Element of the carrier of R^1
( the carrier of I[01],mc100,CM) is Relation-like the carrier of I[01] -defined the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total total quasi_total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
[: the carrier of I[01], the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of I[01], the carrier of R^1:] is set
( the carrier of I[01],mc100,CM) . 0 is complex real ext-real set
j0 is complex real ext-real Element of the carrier of I[01]
mc100 . j0 is complex real ext-real Element of the carrier of R^1
((mc100 . j0),CM) is complex real ext-real Element of the carrier of R^1
( the carrier of I[01],mc100,CM) . 1 is complex real ext-real set
j1 is complex real ext-real Element of the carrier of I[01]
mc100 . j1 is complex real ext-real Element of the carrier of R^1
((mc100 . j1),CM) is complex real ext-real Element of the carrier of R^1
mc100 - CM is Relation-like the carrier of I[01] -defined Function-like total complex-yielding ext-real-valued real-valued set
- CM is complex real ext-real set
(- CM) + mc100 is Relation-like the carrier of I[01] -defined Function-like total complex-yielding ext-real-valued real-valued set
(c100,CM) is complex real ext-real Element of the carrier of R^1
c100 + (- CM) is complex real ext-real set
(c100a,CM) is complex real ext-real Element of the carrier of R^1
c100a + (- CM) is complex real ext-real set
( the carrier of I[01],mc100,CM) is Relation-like the carrier of I[01] -defined the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total total quasi_total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
[: the carrier of I[01], the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of I[01], the carrier of R^1:] is set
( the carrier of I[01],mc100,CM) . 0 is complex real ext-real set
j0 is complex real ext-real Element of the carrier of I[01]
mc100 . j0 is complex real ext-real Element of the carrier of R^1
((mc100 . j0),CM) is complex real ext-real Element of the carrier of R^1
(mc100 . j0) + (- CM) is complex real ext-real set
( the carrier of I[01],mc100,CM) . 1 is complex real ext-real set
j1 is complex real ext-real Element of the carrier of I[01]
mc100 . j1 is complex real ext-real Element of the carrier of R^1
((mc100 . j1),CM) is complex real ext-real Element of the carrier of R^1
(mc100 . j1) + (- CM) is complex real ext-real set
|[1,0,0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
(<*1*> ^ <*0*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[1,0,0]| - (0. (TOP-REAL 3)) is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
|[1,0,0]| - (0. (TOP-REAL 3)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (0. (TOP-REAL 3)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- 1 is non empty complex real integer ext-real non positive negative set
(- 1) * (0. (TOP-REAL 3)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(0. (TOP-REAL 3)) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(0. (TOP-REAL 3)) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[1,0,0]| + (- (0. (TOP-REAL 3))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: (|[1,0,0]|,(- (0. (TOP-REAL 3)))) is Relation-like Function-like set
diffreal .: (|[1,0,0]|,(0. (TOP-REAL 3))) is Relation-like Function-like set
K254(|[1,0,0]|,(0. (TOP-REAL 3))) is Relation-like Function-like set
K254(|[1,0,0]|,(0. (TOP-REAL 3))) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|.(|[1,0,0]| - (0. (TOP-REAL 3))).| is complex real ext-real non negative Element of REAL
sqr (|[1,0,0]| - (0. (TOP-REAL 3))) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((|[1,0,0]| - (0. (TOP-REAL 3))),(|[1,0,0]| - (0. (TOP-REAL 3)))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((|[1,0,0]| - (0. (TOP-REAL 3))),(|[1,0,0]| - (0. (TOP-REAL 3)))) is Relation-like Function-like set
K254((|[1,0,0]| - (0. (TOP-REAL 3))),(|[1,0,0]| - (0. (TOP-REAL 3)))) is Relation-like Function-like set
K254((|[1,0,0]| - (0. (TOP-REAL 3))),(|[1,0,0]| - (0. (TOP-REAL 3)))) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(|[1,0,0]| - (0. (TOP-REAL 3))) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (|[1,0,0]| - (0. (TOP-REAL 3)))) is complex real ext-real Element of REAL
sqrt (Sum (sqr (|[1,0,0]| - (0. (TOP-REAL 3))))) is complex real ext-real Element of REAL
|.|[1,0,0]|.| is complex real ext-real non negative Element of REAL
sqr |[1,0,0]| is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (|[1,0,0]|,|[1,0,0]|) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (|[1,0,0]|,|[1,0,0]|) is Relation-like Function-like set
K254(|[1,0,0]|,|[1,0,0]|) is Relation-like Function-like set
K254(|[1,0,0]|,|[1,0,0]|) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|[1,0,0]| (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr |[1,0,0]|) is complex real ext-real Element of REAL
sqrt (Sum (sqr |[1,0,0]|)) is complex real ext-real Element of REAL
|[1,0,0]| `1 is complex real ext-real Element of REAL
(|[1,0,0]| `1) ^2 is complex real ext-real non negative Element of REAL
(|[1,0,0]| `1) * (|[1,0,0]| `1) is complex real ext-real non negative set
|[1,0,0]| `2 is complex real ext-real Element of REAL
(|[1,0,0]| `2) ^2 is complex real ext-real non negative Element of REAL
(|[1,0,0]| `2) * (|[1,0,0]| `2) is complex real ext-real non negative set
((|[1,0,0]| `1) ^2) + ((|[1,0,0]| `2) ^2) is complex real ext-real non negative Element of REAL
|[1,0,0]| `3 is complex real ext-real Element of REAL
(|[1,0,0]| `3) ^2 is complex real ext-real non negative Element of REAL
(|[1,0,0]| `3) * (|[1,0,0]| `3) is complex real ext-real non negative set
(((|[1,0,0]| `1) ^2) + ((|[1,0,0]| `2) ^2)) + ((|[1,0,0]| `3) ^2) is complex real ext-real non negative Element of REAL
sqrt ((((|[1,0,0]| `1) ^2) + ((|[1,0,0]| `2) ^2)) + ((|[1,0,0]| `3) ^2)) is complex real ext-real non negative Element of REAL
1 ^2 is non empty complex real ext-real positive non negative Element of REAL
1 * 1 is non empty complex ordinal natural real integer ext-real positive non negative set
(1 ^2) + ((|[1,0,0]| `2) ^2) is non empty complex real ext-real positive non negative Element of REAL
((1 ^2) + ((|[1,0,0]| `2) ^2)) + ((|[1,0,0]| `3) ^2) is non empty complex real ext-real positive non negative Element of REAL
sqrt (((1 ^2) + ((|[1,0,0]| `2) ^2)) + ((|[1,0,0]| `3) ^2)) is non empty complex real ext-real positive non negative Element of REAL
Q ^2 is complex real ext-real non negative set
Q * Q is complex ordinal natural real integer ext-real non negative set
(1 ^2) + (Q ^2) is non empty complex real ext-real positive non negative Element of REAL
((1 ^2) + (Q ^2)) + ((|[1,0,0]| `3) ^2) is non empty complex real ext-real positive non negative Element of REAL
sqrt (((1 ^2) + (Q ^2)) + ((|[1,0,0]| `3) ^2)) is non empty complex real ext-real positive non negative Element of REAL
() is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
2 + 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
Tunit_circle (2 + 1) is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (2 + 1)
TOP-REAL (2 + 1) is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
0. (TOP-REAL (2 + 1)) is Relation-like NAT -defined Function-like V32() V39(2 + 1) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (2 + 1))
the carrier of (TOP-REAL (2 + 1)) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (2 + 1)) is Relation-like NAT -defined Function-like V32() V39(2 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (2 + 1))
Tcircle ((0. (TOP-REAL (2 + 1))),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (2 + 1)
Sphere ((0. (TOP-REAL (2 + 1))),1) is functional non empty closed V214( TOP-REAL (2 + 1)) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (2 + 1))
bool the carrier of (TOP-REAL (2 + 1)) is V74() set
(TOP-REAL (2 + 1)) | (Sphere ((0. (TOP-REAL (2 + 1))),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (2 + 1)
the carrier of (Tunit_circle (2 + 1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
|[(- 1),0,0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
(<*(- 1)*> ^ <*0*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- Q is complex real integer ext-real non positive set
|[(- 1),(- Q),(- Q)]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(- Q)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(- Q)] is set
{1,(- Q)} is non empty V195() V196() V197() V198() V199() set
{{1,(- Q)},{1}} is non empty set
{[1,(- Q)]} is Relation-like Function-like non empty set
<*(- 1)*> ^ <*(- Q)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(- 1)*> ^ <*(- Q)*>) ^ <*(- Q)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
c100 is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
- c100 is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
- c100 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- 1 is non empty complex real integer ext-real non positive negative set
(- 1) * c100 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
c100 (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
c100 (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- () is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * () is Relation-like Function-like complex-yielding ext-real-valued real-valued set
() is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
- () is Relation-like Function-like complex-yielding ext-real-valued real-valued set
- 1 is non empty complex real integer ext-real non positive negative set
(- 1) * () is Relation-like Function-like complex-yielding ext-real-valued real-valued set
c100 is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
- c100 is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
- c100 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(- 1) * c100 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
c100 (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
c100 (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- Q is complex real integer ext-real non positive set
|[(- 1),(- Q),(- Q)]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(- Q)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(- Q)] is set
{1,(- Q)} is non empty V195() V196() V197() V198() V199() set
{{1,(- Q)},{1}} is non empty set
{[1,(- Q)]} is Relation-like Function-like non empty set
<*(- 1)*> ^ <*(- Q)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(- 1)*> ^ <*(- Q)*>) ^ <*(- Q)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- () is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * () is Relation-like Function-like complex-yielding ext-real-valued real-valued set
() - () is Relation-like Function-like complex-yielding ext-real-valued real-valued set
() + (- ()) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
|[2,0,0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*2*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,2] is set
{1,2} is non empty V195() V196() V197() V198() V199() V200() set
{{1,2},{1}} is non empty set
{[1,2]} is Relation-like Function-like non empty set
<*2*> ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*2*> ^ <*0*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
c100 is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
mc100 is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
c100 - mc100 is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
c100 - mc100 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- mc100 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * mc100 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
mc100 (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
mc100 (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
c100 + (- mc100) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: (c100,(- mc100)) is Relation-like Function-like set
diffreal .: (c100,mc100) is Relation-like Function-like set
K254(c100,mc100) is Relation-like Function-like set
K254(c100,mc100) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
1 - (- 1) is non empty complex real integer ext-real positive non negative Element of REAL
- (- 1) is non empty complex real integer ext-real positive non negative set
1 + (- (- 1)) is non empty complex real integer ext-real positive non negative set
Q - Q is complex real integer ext-real set
- Q is complex real integer ext-real non positive set
Q + (- Q) is complex real integer ext-real set
|[(1 - (- 1)),(Q - Q),(Q - Q)]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(1 - (- 1))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(1 - (- 1))] is set
{1,(1 - (- 1))} is non empty V195() V196() V197() V198() V199() set
{{1,(1 - (- 1))},{1}} is non empty set
{[1,(1 - (- 1))]} is Relation-like Function-like non empty set
<*(Q - Q)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Q - Q)] is set
{1,(Q - Q)} is non empty V195() V196() V197() V198() V199() set
{{1,(Q - Q)},{1}} is non empty set
{[1,(Q - Q)]} is Relation-like Function-like non empty set
<*(1 - (- 1))*> ^ <*(Q - Q)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(1 - (- 1))*> ^ <*(Q - Q)*>) ^ <*(Q - Q)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
CM is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
CM `1 is complex real ext-real Element of REAL
K382(CM,1) is complex real ext-real Element of REAL
|.CM.| is complex real ext-real non negative Element of REAL
sqr CM is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (CM,CM) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (CM,CM) is Relation-like Function-like set
K254(CM,CM) is Relation-like Function-like set
K254(CM,CM) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
CM (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr CM) is complex real ext-real Element of REAL
sqrt (Sum (sqr CM)) is complex real ext-real Element of REAL
Arg CM is complex real ext-real Element of REAL
euc2cpx CM is complex Element of COMPLEX
CM `2 is complex real ext-real Element of REAL
K382(CM,2) is complex real ext-real Element of REAL
(CM `2) * <i> is complex set
(CM `1) + ((CM `2) * <i>) is complex set
Arg (euc2cpx CM) is complex real ext-real Element of REAL
cos (Arg CM) is complex real ext-real Element of REAL
|.CM.| * (cos (Arg CM)) is complex real ext-real Element of REAL
sin (Arg CM) is complex real ext-real Element of REAL
|.CM.| * (sin (Arg CM)) is complex real ext-real Element of REAL
|.(euc2cpx CM).| is complex real ext-real Element of REAL
Re (euc2cpx CM) is complex real ext-real Element of REAL
(Re (euc2cpx CM)) ^2 is complex real ext-real non negative Element of REAL
(Re (euc2cpx CM)) * (Re (euc2cpx CM)) is complex real ext-real non negative set
Im (euc2cpx CM) is complex real ext-real Element of REAL
(Im (euc2cpx CM)) ^2 is complex real ext-real non negative Element of REAL
(Im (euc2cpx CM)) * (Im (euc2cpx CM)) is complex real ext-real non negative set
((Re (euc2cpx CM)) ^2) + ((Im (euc2cpx CM)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (euc2cpx CM)) ^2) + ((Im (euc2cpx CM)) ^2)) is complex real ext-real non negative Element of REAL
cos (Arg (euc2cpx CM)) is complex real ext-real Element of REAL
|.(euc2cpx CM).| * (cos (Arg (euc2cpx CM))) is complex real ext-real Element of REAL
sin (Arg (euc2cpx CM)) is complex real ext-real Element of REAL
|.(euc2cpx CM).| * (sin (Arg (euc2cpx CM))) is complex real ext-real Element of REAL
(|.(euc2cpx CM).| * (sin (Arg (euc2cpx CM)))) * <i> is complex set
(|.(euc2cpx CM).| * (cos (Arg (euc2cpx CM)))) + ((|.(euc2cpx CM).| * (sin (Arg (euc2cpx CM)))) * <i>) is complex set
CM is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
|.CM.| is complex real ext-real non negative Element of REAL
sqr CM is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (CM,CM) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (CM,CM) is Relation-like Function-like set
K254(CM,CM) is Relation-like Function-like set
K254(CM,CM) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
CM (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr CM) is complex real ext-real Element of REAL
sqrt (Sum (sqr CM)) is complex real ext-real Element of REAL
Arg CM is complex real ext-real Element of REAL
euc2cpx CM is complex Element of COMPLEX
CM `1 is complex real ext-real Element of REAL
K382(CM,1) is complex real ext-real Element of REAL
CM `2 is complex real ext-real Element of REAL
K382(CM,2) is complex real ext-real Element of REAL
(CM `2) * <i> is complex set
(CM `1) + ((CM `2) * <i>) is complex set
Arg (euc2cpx CM) is complex real ext-real Element of REAL
cos (Arg CM) is complex real ext-real Element of REAL
|.CM.| * (cos (Arg CM)) is complex real ext-real Element of REAL
sin (Arg CM) is complex real ext-real Element of REAL
|.CM.| * (sin (Arg CM)) is complex real ext-real Element of REAL
(|.CM.| * (sin (Arg CM))) * <i> is complex set
(|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>) is complex set
cpx2euc ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Re ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)) is complex real ext-real Element of REAL
Im ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)) is complex real ext-real Element of REAL
|[(Re ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>))),(Im ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(Re ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Re ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))] is set
{1,(Re ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))} is non empty V195() V196() V197() set
{{1,(Re ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))},{1}} is non empty set
{[1,(Re ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))]} is Relation-like Function-like non empty set
<*(Im ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Im ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))] is set
{1,(Im ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))} is non empty V195() V196() V197() set
{{1,(Im ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))},{1}} is non empty set
{[1,(Im ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))]} is Relation-like Function-like non empty set
<*(Re ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))*> ^ <*(Im ((|.CM.| * (cos (Arg CM))) + ((|.CM.| * (sin (Arg CM))) * <i>)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|.(euc2cpx CM).| is complex real ext-real Element of REAL
Re (euc2cpx CM) is complex real ext-real Element of REAL
(Re (euc2cpx CM)) ^2 is complex real ext-real non negative Element of REAL
(Re (euc2cpx CM)) * (Re (euc2cpx CM)) is complex real ext-real non negative set
Im (euc2cpx CM) is complex real ext-real Element of REAL
(Im (euc2cpx CM)) ^2 is complex real ext-real non negative Element of REAL
(Im (euc2cpx CM)) * (Im (euc2cpx CM)) is complex real ext-real non negative set
((Re (euc2cpx CM)) ^2) + ((Im (euc2cpx CM)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (euc2cpx CM)) ^2) + ((Im (euc2cpx CM)) ^2)) is complex real ext-real non negative Element of REAL
cos (Arg (euc2cpx CM)) is complex real ext-real Element of REAL
|.(euc2cpx CM).| * (cos (Arg (euc2cpx CM))) is complex real ext-real Element of REAL
sin (Arg (euc2cpx CM)) is complex real ext-real Element of REAL
|.(euc2cpx CM).| * (sin (Arg (euc2cpx CM))) is complex real ext-real Element of REAL
(|.(euc2cpx CM).| * (sin (Arg (euc2cpx CM)))) * <i> is complex set
(|.(euc2cpx CM).| * (cos (Arg (euc2cpx CM)))) + ((|.(euc2cpx CM).| * (sin (Arg (euc2cpx CM)))) * <i>) is complex set
CM is complex real integer ext-real set
(2 * PI) * CM is complex real ext-real Element of REAL
SM is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
|.SM.| is complex real ext-real non negative Element of REAL
sqr SM is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (SM,SM) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (SM,SM) is Relation-like Function-like set
K254(SM,SM) is Relation-like Function-like set
K254(SM,SM) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
SM (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr SM) is complex real ext-real Element of REAL
sqrt (Sum (sqr SM)) is complex real ext-real Element of REAL
r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
|.r.| is complex real ext-real non negative Element of REAL
sqr r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (r,r) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (r,r) is Relation-like Function-like set
K254(r,r) is Relation-like Function-like set
K254(r,r) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
r (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr r) is complex real ext-real Element of REAL
sqrt (Sum (sqr r)) is complex real ext-real Element of REAL
Arg SM is complex real ext-real Element of REAL
euc2cpx SM is complex Element of COMPLEX
SM `1 is complex real ext-real Element of REAL
K382(SM,1) is complex real ext-real Element of REAL
SM `2 is complex real ext-real Element of REAL
K382(SM,2) is complex real ext-real Element of REAL
(SM `2) * <i> is complex set
(SM `1) + ((SM `2) * <i>) is complex set
Arg (euc2cpx SM) is complex real ext-real Element of REAL
Arg r is complex real ext-real Element of REAL
euc2cpx r is complex Element of COMPLEX
r `1 is complex real ext-real Element of REAL
K382(r,1) is complex real ext-real Element of REAL
r `2 is complex real ext-real Element of REAL
K382(r,2) is complex real ext-real Element of REAL
(r `2) * <i> is complex set
(r `1) + ((r `2) * <i>) is complex set
Arg (euc2cpx r) is complex real ext-real Element of REAL
(Arg r) + ((2 * PI) * CM) is complex real ext-real Element of REAL
|.(euc2cpx SM).| is complex real ext-real Element of REAL
Re (euc2cpx SM) is complex real ext-real Element of REAL
(Re (euc2cpx SM)) ^2 is complex real ext-real non negative Element of REAL
(Re (euc2cpx SM)) * (Re (euc2cpx SM)) is complex real ext-real non negative set
Im (euc2cpx SM) is complex real ext-real Element of REAL
(Im (euc2cpx SM)) ^2 is complex real ext-real non negative Element of REAL
(Im (euc2cpx SM)) * (Im (euc2cpx SM)) is complex real ext-real non negative set
((Re (euc2cpx SM)) ^2) + ((Im (euc2cpx SM)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (euc2cpx SM)) ^2) + ((Im (euc2cpx SM)) ^2)) is complex real ext-real non negative Element of REAL
|.(euc2cpx r).| is complex real ext-real Element of REAL
Re (euc2cpx r) is complex real ext-real Element of REAL
(Re (euc2cpx r)) ^2 is complex real ext-real non negative Element of REAL
(Re (euc2cpx r)) * (Re (euc2cpx r)) is complex real ext-real non negative set
Im (euc2cpx r) is complex real ext-real Element of REAL
(Im (euc2cpx r)) ^2 is complex real ext-real non negative Element of REAL
(Im (euc2cpx r)) * (Im (euc2cpx r)) is complex real ext-real non negative set
((Re (euc2cpx r)) ^2) + ((Im (euc2cpx r)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (euc2cpx r)) ^2) + ((Im (euc2cpx r)) ^2)) is complex real ext-real non negative Element of REAL
SM is complex real ext-real set
CircleMap . SM is Relation-like Function-like complex-yielding ext-real-valued real-valued set
frac SM is complex real ext-real Element of REAL
[\SM/] is complex real integer ext-real set
SM - [\SM/] is complex real ext-real set
- [\SM/] is complex real integer ext-real set
SM + (- [\SM/]) is complex real ext-real set
(2 * PI) * (frac SM) is complex real ext-real Element of REAL
r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Arg r is complex real ext-real Element of REAL
euc2cpx r is complex Element of COMPLEX
r `1 is complex real ext-real Element of REAL
K382(r,1) is complex real ext-real Element of REAL
r `2 is complex real ext-real Element of REAL
K382(r,2) is complex real ext-real Element of REAL
(r `2) * <i> is complex set
(r `1) + ((r `2) * <i>) is complex set
Arg (euc2cpx r) is complex real ext-real Element of REAL
R^1 SM is complex real ext-real Element of the carrier of K708()
CircleMap . (R^1 SM) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
|.(euc2cpx r).| is complex real ext-real Element of REAL
Re (euc2cpx r) is complex real ext-real Element of REAL
(Re (euc2cpx r)) ^2 is complex real ext-real non negative Element of REAL
(Re (euc2cpx r)) * (Re (euc2cpx r)) is complex real ext-real non negative set
Im (euc2cpx r) is complex real ext-real Element of REAL
(Im (euc2cpx r)) ^2 is complex real ext-real non negative Element of REAL
(Im (euc2cpx r)) * (Im (euc2cpx r)) is complex real ext-real non negative set
((Re (euc2cpx r)) ^2) + ((Im (euc2cpx r)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (euc2cpx r)) ^2) + ((Im (euc2cpx r)) ^2)) is complex real ext-real non negative Element of REAL
|.r.| is complex real ext-real non negative Element of REAL
sqr r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (r,r) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (r,r) is Relation-like Function-like set
K254(r,r) is Relation-like Function-like set
K254(r,r) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
r (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr r) is complex real ext-real Element of REAL
sqrt (Sum (sqr r)) is complex real ext-real Element of REAL
e is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
|.e.| is complex real ext-real non negative Element of REAL
sqr e is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (e,e) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (e,e) is Relation-like Function-like set
K254(e,e) is Relation-like Function-like set
K254(e,e) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
e (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr e) is complex real ext-real Element of REAL
sqrt (Sum (sqr e)) is complex real ext-real Element of REAL
(2 * PI) * Q is complex real ext-real non negative Element of REAL
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * SM is complex real ext-real Element of REAL
cos ((2 * PI) * SM) is complex real ext-real Element of REAL
sin ((2 * PI) * SM) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM))]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(cos ((2 * PI) * SM))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos ((2 * PI) * SM))] is set
{1,(cos ((2 * PI) * SM))} is non empty V195() V196() V197() set
{{1,(cos ((2 * PI) * SM))},{1}} is non empty set
{[1,(cos ((2 * PI) * SM))]} is Relation-like Function-like non empty set
<*(sin ((2 * PI) * SM))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin ((2 * PI) * SM))] is set
{1,(sin ((2 * PI) * SM))} is non empty V195() V196() V197() set
{{1,(sin ((2 * PI) * SM))},{1}} is non empty set
{[1,(sin ((2 * PI) * SM))]} is Relation-like Function-like non empty set
<*(cos ((2 * PI) * SM))*> ^ <*(sin ((2 * PI) * SM))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM))]| `1 is complex real ext-real Element of REAL
K382(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM))]|,1) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM))]| `2 is complex real ext-real Element of REAL
K382(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM))]|,2) is complex real ext-real Element of REAL
(2 * PI) * (- [\SM/]) is complex real ext-real Element of REAL
((2 * PI) * SM) + ((2 * PI) * (- [\SM/])) is complex real ext-real Element of REAL
cos ((2 * PI) * (frac SM)) is complex real ext-real Element of REAL
sin ((2 * PI) * (frac SM)) is complex real ext-real Element of REAL
|.(euc2cpx r).| * (cos ((2 * PI) * (frac SM))) is complex real ext-real Element of REAL
|.(euc2cpx r).| * (sin ((2 * PI) * (frac SM))) is complex real ext-real Element of REAL
(|.(euc2cpx r).| * (sin ((2 * PI) * (frac SM)))) * <i> is complex set
(|.(euc2cpx r).| * (cos ((2 * PI) * (frac SM)))) + ((|.(euc2cpx r).| * (sin ((2 * PI) * (frac SM)))) * <i>) is complex set
Euclid 3 is non empty strict Reflexive discerning symmetric triangle complex-functions-membered real-functions-membered MetrStruct
REAL 3 is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered M29( REAL )
K513(3,REAL) is functional non empty FinSequence-membered M29( REAL )
Pitag_dist 3 is Relation-like [:(REAL 3),(REAL 3):] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:(REAL 3),(REAL 3):],REAL:]
[:(REAL 3),(REAL 3):] is Relation-like set
[:[:(REAL 3),(REAL 3):],REAL:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:[:(REAL 3),(REAL 3):],REAL:] is set
MetrStruct(# (REAL 3),(Pitag_dist 3) #) is strict MetrStruct
the carrier of (Euclid 3) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
SM is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
r is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
SM `1 is complex real ext-real Element of REAL
r `1 is complex real ext-real Element of REAL
(SM `1) - (r `1) is complex real ext-real Element of REAL
- (r `1) is complex real ext-real set
(SM `1) + (- (r `1)) is complex real ext-real set
((SM `1) - (r `1)) ^2 is complex real ext-real non negative Element of REAL
((SM `1) - (r `1)) * ((SM `1) - (r `1)) is complex real ext-real non negative set
SM `2 is complex real ext-real Element of REAL
r `2 is complex real ext-real Element of REAL
(SM `2) - (r `2) is complex real ext-real Element of REAL
- (r `2) is complex real ext-real set
(SM `2) + (- (r `2)) is complex real ext-real set
((SM `2) - (r `2)) ^2 is complex real ext-real non negative Element of REAL
((SM `2) - (r `2)) * ((SM `2) - (r `2)) is complex real ext-real non negative set
(((SM `1) - (r `1)) ^2) + (((SM `2) - (r `2)) ^2) is complex real ext-real non negative Element of REAL
SM `3 is complex real ext-real Element of REAL
r `3 is complex real ext-real Element of REAL
(SM `3) - (r `3) is complex real ext-real Element of REAL
- (r `3) is complex real ext-real set
(SM `3) + (- (r `3)) is complex real ext-real set
((SM `3) - (r `3)) ^2 is complex real ext-real non negative Element of REAL
((SM `3) - (r `3)) * ((SM `3) - (r `3)) is complex real ext-real non negative set
((((SM `1) - (r `1)) ^2) + (((SM `2) - (r `2)) ^2)) + (((SM `3) - (r `3)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((((SM `1) - (r `1)) ^2) + (((SM `2) - (r `2)) ^2)) + (((SM `3) - (r `3)) ^2)) is complex real ext-real non negative Element of REAL
p is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Euclid 3)
f is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Euclid 3)
(Pitag_dist 3) . (p,f) is set
|[(SM `1),(SM `2),(SM `3)]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(SM `1)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(SM `1)] is set
{1,(SM `1)} is non empty V195() V196() V197() set
{{1,(SM `1)},{1}} is non empty set
{[1,(SM `1)]} is Relation-like Function-like non empty set
<*(SM `2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(SM `2)] is set
{1,(SM `2)} is non empty V195() V196() V197() set
{{1,(SM `2)},{1}} is non empty set
{[1,(SM `2)]} is Relation-like Function-like non empty set
<*(SM `1)*> ^ <*(SM `2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(SM `3)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(SM `3)] is set
{1,(SM `3)} is non empty V195() V196() V197() set
{{1,(SM `3)},{1}} is non empty set
{[1,(SM `3)]} is Relation-like Function-like non empty set
(<*(SM `1)*> ^ <*(SM `2)*>) ^ <*(SM `3)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(r `1),(r `2),(r `3)*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(r `1)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(r `1)] is set
{1,(r `1)} is non empty V195() V196() V197() set
{{1,(r `1)},{1}} is non empty set
{[1,(r `1)]} is Relation-like Function-like non empty set
<*(r `2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(r `2)] is set
{1,(r `2)} is non empty V195() V196() V197() set
{{1,(r `2)},{1}} is non empty set
{[1,(r `2)]} is Relation-like Function-like non empty set
<*(r `1)*> ^ <*(r `2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(r `3)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(r `3)] is set
{1,(r `3)} is non empty V195() V196() V197() set
{{1,(r `3)},{1}} is non empty set
{[1,(r `3)]} is Relation-like Function-like non empty set
(<*(r `1)*> ^ <*(r `2)*>) ^ <*(r `3)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
e is Relation-like NAT -defined REAL -valued Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of REAL 3
f1 is Relation-like NAT -defined REAL -valued Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of REAL 3
e - f1 is Relation-like NAT -defined REAL -valued Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of REAL 3
- f1 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * f1 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
f1 (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
f1 (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
e + (- f1) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: (e,(- f1)) is Relation-like Function-like set
K254(e,(- f1)) is Relation-like Function-like set
K254(e,(- f1)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
diffreal .: (e,f1) is Relation-like Function-like set
K254(e,f1) is Relation-like Function-like set
K254(e,f1) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
diffreal . ((SM `1),(r `1)) is complex real ext-real Element of REAL
diffreal . ((SM `2),(r `2)) is complex real ext-real Element of REAL
diffreal . ((SM `3),(r `3)) is complex real ext-real Element of REAL
<*(diffreal . ((SM `1),(r `1))),(diffreal . ((SM `2),(r `2))),(diffreal . ((SM `3),(r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(diffreal . ((SM `1),(r `1)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(diffreal . ((SM `1),(r `1)))] is set
{1,(diffreal . ((SM `1),(r `1)))} is non empty V195() V196() V197() set
{{1,(diffreal . ((SM `1),(r `1)))},{1}} is non empty set
{[1,(diffreal . ((SM `1),(r `1)))]} is Relation-like Function-like non empty set
<*(diffreal . ((SM `2),(r `2)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(diffreal . ((SM `2),(r `2)))] is set
{1,(diffreal . ((SM `2),(r `2)))} is non empty V195() V196() V197() set
{{1,(diffreal . ((SM `2),(r `2)))},{1}} is non empty set
{[1,(diffreal . ((SM `2),(r `2)))]} is Relation-like Function-like non empty set
<*(diffreal . ((SM `1),(r `1)))*> ^ <*(diffreal . ((SM `2),(r `2)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(diffreal . ((SM `3),(r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(diffreal . ((SM `3),(r `3)))] is set
{1,(diffreal . ((SM `3),(r `3)))} is non empty V195() V196() V197() set
{{1,(diffreal . ((SM `3),(r `3)))},{1}} is non empty set
{[1,(diffreal . ((SM `3),(r `3)))]} is Relation-like Function-like non empty set
(<*(diffreal . ((SM `1),(r `1)))*> ^ <*(diffreal . ((SM `2),(r `2)))*>) ^ <*(diffreal . ((SM `3),(r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((SM `1) - (r `1)),(diffreal . ((SM `2),(r `2))),(diffreal . ((SM `3),(r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((SM `1) - (r `1))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((SM `1) - (r `1))] is set
{1,((SM `1) - (r `1))} is non empty V195() V196() V197() set
{{1,((SM `1) - (r `1))},{1}} is non empty set
{[1,((SM `1) - (r `1))]} is Relation-like Function-like non empty set
<*((SM `1) - (r `1))*> ^ <*(diffreal . ((SM `2),(r `2)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*((SM `1) - (r `1))*> ^ <*(diffreal . ((SM `2),(r `2)))*>) ^ <*(diffreal . ((SM `3),(r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((SM `1) - (r `1)),((SM `2) - (r `2)),(diffreal . ((SM `3),(r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((SM `2) - (r `2))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((SM `2) - (r `2))] is set
{1,((SM `2) - (r `2))} is non empty V195() V196() V197() set
{{1,((SM `2) - (r `2))},{1}} is non empty set
{[1,((SM `2) - (r `2))]} is Relation-like Function-like non empty set
<*((SM `1) - (r `1))*> ^ <*((SM `2) - (r `2))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*((SM `1) - (r `1))*> ^ <*((SM `2) - (r `2))*>) ^ <*(diffreal . ((SM `3),(r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((SM `1) - (r `1)),((SM `2) - (r `2)),((SM `3) - (r `3))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((SM `3) - (r `3))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((SM `3) - (r `3))] is set
{1,((SM `3) - (r `3))} is non empty V195() V196() V197() set
{{1,((SM `3) - (r `3))},{1}} is non empty set
{[1,((SM `3) - (r `3))]} is Relation-like Function-like non empty set
(<*((SM `1) - (r `1))*> ^ <*((SM `2) - (r `2))*>) ^ <*((SM `3) - (r `3))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
abs (e - f1) is Relation-like NAT -defined REAL -valued Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of K513(3,REAL)
(e - f1) (#) absreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
absreal . ((SM `1) - (r `1)) is complex real ext-real Element of REAL
absreal . ((SM `2) - (r `2)) is complex real ext-real Element of REAL
absreal . ((SM `3) - (r `3)) is complex real ext-real Element of REAL
<*(absreal . ((SM `1) - (r `1))),(absreal . ((SM `2) - (r `2))),(absreal . ((SM `3) - (r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(absreal . ((SM `1) - (r `1)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(absreal . ((SM `1) - (r `1)))] is set
{1,(absreal . ((SM `1) - (r `1)))} is non empty V195() V196() V197() set
{{1,(absreal . ((SM `1) - (r `1)))},{1}} is non empty set
{[1,(absreal . ((SM `1) - (r `1)))]} is Relation-like Function-like non empty set
<*(absreal . ((SM `2) - (r `2)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(absreal . ((SM `2) - (r `2)))] is set
{1,(absreal . ((SM `2) - (r `2)))} is non empty V195() V196() V197() set
{{1,(absreal . ((SM `2) - (r `2)))},{1}} is non empty set
{[1,(absreal . ((SM `2) - (r `2)))]} is Relation-like Function-like non empty set
<*(absreal . ((SM `1) - (r `1)))*> ^ <*(absreal . ((SM `2) - (r `2)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(absreal . ((SM `3) - (r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(absreal . ((SM `3) - (r `3)))] is set
{1,(absreal . ((SM `3) - (r `3)))} is non empty V195() V196() V197() set
{{1,(absreal . ((SM `3) - (r `3)))},{1}} is non empty set
{[1,(absreal . ((SM `3) - (r `3)))]} is Relation-like Function-like non empty set
(<*(absreal . ((SM `1) - (r `1)))*> ^ <*(absreal . ((SM `2) - (r `2)))*>) ^ <*(absreal . ((SM `3) - (r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
abs ((SM `1) - (r `1)) is complex real ext-real Element of REAL
Re ((SM `1) - (r `1)) is complex real ext-real Element of REAL
(Re ((SM `1) - (r `1))) ^2 is complex real ext-real non negative Element of REAL
(Re ((SM `1) - (r `1))) * (Re ((SM `1) - (r `1))) is complex real ext-real non negative set
Im ((SM `1) - (r `1)) is complex real ext-real Element of REAL
(Im ((SM `1) - (r `1))) ^2 is complex real ext-real non negative Element of REAL
(Im ((SM `1) - (r `1))) * (Im ((SM `1) - (r `1))) is complex real ext-real non negative set
((Re ((SM `1) - (r `1))) ^2) + ((Im ((SM `1) - (r `1))) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re ((SM `1) - (r `1))) ^2) + ((Im ((SM `1) - (r `1))) ^2)) is complex real ext-real non negative Element of REAL
<*(abs ((SM `1) - (r `1))),(absreal . ((SM `2) - (r `2))),(absreal . ((SM `3) - (r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(abs ((SM `1) - (r `1)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(abs ((SM `1) - (r `1)))] is set
{1,(abs ((SM `1) - (r `1)))} is non empty V195() V196() V197() set
{{1,(abs ((SM `1) - (r `1)))},{1}} is non empty set
{[1,(abs ((SM `1) - (r `1)))]} is Relation-like Function-like non empty set
<*(abs ((SM `1) - (r `1)))*> ^ <*(absreal . ((SM `2) - (r `2)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(abs ((SM `1) - (r `1)))*> ^ <*(absreal . ((SM `2) - (r `2)))*>) ^ <*(absreal . ((SM `3) - (r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
abs ((SM `2) - (r `2)) is complex real ext-real Element of REAL
Re ((SM `2) - (r `2)) is complex real ext-real Element of REAL
(Re ((SM `2) - (r `2))) ^2 is complex real ext-real non negative Element of REAL
(Re ((SM `2) - (r `2))) * (Re ((SM `2) - (r `2))) is complex real ext-real non negative set
Im ((SM `2) - (r `2)) is complex real ext-real Element of REAL
(Im ((SM `2) - (r `2))) ^2 is complex real ext-real non negative Element of REAL
(Im ((SM `2) - (r `2))) * (Im ((SM `2) - (r `2))) is complex real ext-real non negative set
((Re ((SM `2) - (r `2))) ^2) + ((Im ((SM `2) - (r `2))) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re ((SM `2) - (r `2))) ^2) + ((Im ((SM `2) - (r `2))) ^2)) is complex real ext-real non negative Element of REAL
<*(abs ((SM `1) - (r `1))),(abs ((SM `2) - (r `2))),(absreal . ((SM `3) - (r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(abs ((SM `2) - (r `2)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(abs ((SM `2) - (r `2)))] is set
{1,(abs ((SM `2) - (r `2)))} is non empty V195() V196() V197() set
{{1,(abs ((SM `2) - (r `2)))},{1}} is non empty set
{[1,(abs ((SM `2) - (r `2)))]} is Relation-like Function-like non empty set
<*(abs ((SM `1) - (r `1)))*> ^ <*(abs ((SM `2) - (r `2)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(abs ((SM `1) - (r `1)))*> ^ <*(abs ((SM `2) - (r `2)))*>) ^ <*(absreal . ((SM `3) - (r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
abs ((SM `3) - (r `3)) is complex real ext-real Element of REAL
Re ((SM `3) - (r `3)) is complex real ext-real Element of REAL
(Re ((SM `3) - (r `3))) ^2 is complex real ext-real non negative Element of REAL
(Re ((SM `3) - (r `3))) * (Re ((SM `3) - (r `3))) is complex real ext-real non negative set
Im ((SM `3) - (r `3)) is complex real ext-real Element of REAL
(Im ((SM `3) - (r `3))) ^2 is complex real ext-real non negative Element of REAL
(Im ((SM `3) - (r `3))) * (Im ((SM `3) - (r `3))) is complex real ext-real non negative set
((Re ((SM `3) - (r `3))) ^2) + ((Im ((SM `3) - (r `3))) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re ((SM `3) - (r `3))) ^2) + ((Im ((SM `3) - (r `3))) ^2)) is complex real ext-real non negative Element of REAL
<*(abs ((SM `1) - (r `1))),(abs ((SM `2) - (r `2))),(abs ((SM `3) - (r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(abs ((SM `3) - (r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(abs ((SM `3) - (r `3)))] is set
{1,(abs ((SM `3) - (r `3)))} is non empty V195() V196() V197() set
{{1,(abs ((SM `3) - (r `3)))},{1}} is non empty set
{[1,(abs ((SM `3) - (r `3)))]} is Relation-like Function-like non empty set
(<*(abs ((SM `1) - (r `1)))*> ^ <*(abs ((SM `2) - (r `2)))*>) ^ <*(abs ((SM `3) - (r `3)))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
sqr (abs (e - f1)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of K513(3,REAL)
mlt ((abs (e - f1)),(abs (e - f1))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((abs (e - f1)),(abs (e - f1))) is Relation-like Function-like set
K254((abs (e - f1)),(abs (e - f1))) is Relation-like Function-like set
K254((abs (e - f1)),(abs (e - f1))) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(abs (e - f1)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
sqrreal . (abs ((SM `1) - (r `1))) is complex real ext-real Element of REAL
sqrreal . (abs ((SM `2) - (r `2))) is complex real ext-real Element of REAL
sqrreal . (abs ((SM `3) - (r `3))) is complex real ext-real Element of REAL
<*(sqrreal . (abs ((SM `1) - (r `1)))),(sqrreal . (abs ((SM `2) - (r `2)))),(sqrreal . (abs ((SM `3) - (r `3))))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(sqrreal . (abs ((SM `1) - (r `1))))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sqrreal . (abs ((SM `1) - (r `1))))] is set
{1,(sqrreal . (abs ((SM `1) - (r `1))))} is non empty V195() V196() V197() set
{{1,(sqrreal . (abs ((SM `1) - (r `1))))},{1}} is non empty set
{[1,(sqrreal . (abs ((SM `1) - (r `1))))]} is Relation-like Function-like non empty set
<*(sqrreal . (abs ((SM `2) - (r `2))))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sqrreal . (abs ((SM `2) - (r `2))))] is set
{1,(sqrreal . (abs ((SM `2) - (r `2))))} is non empty V195() V196() V197() set
{{1,(sqrreal . (abs ((SM `2) - (r `2))))},{1}} is non empty set
{[1,(sqrreal . (abs ((SM `2) - (r `2))))]} is Relation-like Function-like non empty set
<*(sqrreal . (abs ((SM `1) - (r `1))))*> ^ <*(sqrreal . (abs ((SM `2) - (r `2))))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(sqrreal . (abs ((SM `3) - (r `3))))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sqrreal . (abs ((SM `3) - (r `3))))] is set
{1,(sqrreal . (abs ((SM `3) - (r `3))))} is non empty V195() V196() V197() set
{{1,(sqrreal . (abs ((SM `3) - (r `3))))},{1}} is non empty set
{[1,(sqrreal . (abs ((SM `3) - (r `3))))]} is Relation-like Function-like non empty set
(<*(sqrreal . (abs ((SM `1) - (r `1))))*> ^ <*(sqrreal . (abs ((SM `2) - (r `2))))*>) ^ <*(sqrreal . (abs ((SM `3) - (r `3))))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(abs ((SM `1) - (r `1))) ^2 is complex real ext-real non negative Element of REAL
(abs ((SM `1) - (r `1))) * (abs ((SM `1) - (r `1))) is complex real ext-real non negative set
<*((abs ((SM `1) - (r `1))) ^2),(sqrreal . (abs ((SM `2) - (r `2)))),(sqrreal . (abs ((SM `3) - (r `3))))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((abs ((SM `1) - (r `1))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((abs ((SM `1) - (r `1))) ^2)] is set
{1,((abs ((SM `1) - (r `1))) ^2)} is non empty V195() V196() V197() set
{{1,((abs ((SM `1) - (r `1))) ^2)},{1}} is non empty set
{[1,((abs ((SM `1) - (r `1))) ^2)]} is Relation-like Function-like non empty set
<*((abs ((SM `1) - (r `1))) ^2)*> ^ <*(sqrreal . (abs ((SM `2) - (r `2))))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*((abs ((SM `1) - (r `1))) ^2)*> ^ <*(sqrreal . (abs ((SM `2) - (r `2))))*>) ^ <*(sqrreal . (abs ((SM `3) - (r `3))))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(abs ((SM `2) - (r `2))) ^2 is complex real ext-real non negative Element of REAL
(abs ((SM `2) - (r `2))) * (abs ((SM `2) - (r `2))) is complex real ext-real non negative set
<*((abs ((SM `1) - (r `1))) ^2),((abs ((SM `2) - (r `2))) ^2),(sqrreal . (abs ((SM `3) - (r `3))))*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((abs ((SM `2) - (r `2))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((abs ((SM `2) - (r `2))) ^2)] is set
{1,((abs ((SM `2) - (r `2))) ^2)} is non empty V195() V196() V197() set
{{1,((abs ((SM `2) - (r `2))) ^2)},{1}} is non empty set
{[1,((abs ((SM `2) - (r `2))) ^2)]} is Relation-like Function-like non empty set
<*((abs ((SM `1) - (r `1))) ^2)*> ^ <*((abs ((SM `2) - (r `2))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*((abs ((SM `1) - (r `1))) ^2)*> ^ <*((abs ((SM `2) - (r `2))) ^2)*>) ^ <*(sqrreal . (abs ((SM `3) - (r `3))))*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(abs ((SM `3) - (r `3))) ^2 is complex real ext-real non negative Element of REAL
(abs ((SM `3) - (r `3))) * (abs ((SM `3) - (r `3))) is complex real ext-real non negative set
<*((abs ((SM `1) - (r `1))) ^2),((abs ((SM `2) - (r `2))) ^2),((abs ((SM `3) - (r `3))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*((abs ((SM `3) - (r `3))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((abs ((SM `3) - (r `3))) ^2)] is set
{1,((abs ((SM `3) - (r `3))) ^2)} is non empty V195() V196() V197() set
{{1,((abs ((SM `3) - (r `3))) ^2)},{1}} is non empty set
{[1,((abs ((SM `3) - (r `3))) ^2)]} is Relation-like Function-like non empty set
(<*((abs ((SM `1) - (r `1))) ^2)*> ^ <*((abs ((SM `2) - (r `2))) ^2)*>) ^ <*((abs ((SM `3) - (r `3))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(((SM `1) - (r `1)) ^2),((abs ((SM `2) - (r `2))) ^2),((abs ((SM `3) - (r `3))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(((SM `1) - (r `1)) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(((SM `1) - (r `1)) ^2)] is set
{1,(((SM `1) - (r `1)) ^2)} is non empty V195() V196() V197() set
{{1,(((SM `1) - (r `1)) ^2)},{1}} is non empty set
{[1,(((SM `1) - (r `1)) ^2)]} is Relation-like Function-like non empty set
<*(((SM `1) - (r `1)) ^2)*> ^ <*((abs ((SM `2) - (r `2))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(((SM `1) - (r `1)) ^2)*> ^ <*((abs ((SM `2) - (r `2))) ^2)*>) ^ <*((abs ((SM `3) - (r `3))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(((SM `1) - (r `1)) ^2),(((SM `2) - (r `2)) ^2),((abs ((SM `3) - (r `3))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(((SM `2) - (r `2)) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(((SM `2) - (r `2)) ^2)] is set
{1,(((SM `2) - (r `2)) ^2)} is non empty V195() V196() V197() set
{{1,(((SM `2) - (r `2)) ^2)},{1}} is non empty set
{[1,(((SM `2) - (r `2)) ^2)]} is Relation-like Function-like non empty set
<*(((SM `1) - (r `1)) ^2)*> ^ <*(((SM `2) - (r `2)) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(((SM `1) - (r `1)) ^2)*> ^ <*(((SM `2) - (r `2)) ^2)*>) ^ <*((abs ((SM `3) - (r `3))) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(((SM `1) - (r `1)) ^2),(((SM `2) - (r `2)) ^2),(((SM `3) - (r `3)) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(((SM `3) - (r `3)) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(((SM `3) - (r `3)) ^2)] is set
{1,(((SM `3) - (r `3)) ^2)} is non empty V195() V196() V197() set
{{1,(((SM `3) - (r `3)) ^2)},{1}} is non empty set
{[1,(((SM `3) - (r `3)) ^2)]} is Relation-like Function-like non empty set
(<*(((SM `1) - (r `1)) ^2)*> ^ <*(((SM `2) - (r `2)) ^2)*>) ^ <*(((SM `3) - (r `3)) ^2)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|.(e - f1).| is complex real ext-real non negative Element of REAL
sqr (e - f1) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((e - f1),(e - f1)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((e - f1),(e - f1)) is Relation-like Function-like set
K254((e - f1),(e - f1)) is Relation-like Function-like set
K254((e - f1),(e - f1)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(e - f1) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (e - f1)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (e - f1))) is complex real ext-real Element of REAL
Sum (sqr (abs (e - f1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr (abs (e - f1)))) is complex real ext-real Element of REAL
sqrt 2 is non empty complex real ext-real positive non negative Element of REAL
SM is complex real ext-real set
SM / (sqrt 2) is complex real ext-real Element of COMPLEX
(sqrt 2) " is non empty complex real ext-real positive non negative set
SM * ((sqrt 2) ") is complex real ext-real set
r is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
r `3 is complex real ext-real Element of REAL
r `1 is complex real ext-real Element of REAL
(r `1) - (SM / (sqrt 2)) is complex real ext-real Element of REAL
- (SM / (sqrt 2)) is complex real ext-real set
(r `1) + (- (SM / (sqrt 2))) is complex real ext-real set
(r `1) + (SM / (sqrt 2)) is complex real ext-real Element of REAL
].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[ is V195() V196() V197() open Element of bool REAL
r `2 is complex real ext-real Element of REAL
(r `2) - (SM / (sqrt 2)) is complex real ext-real Element of REAL
(r `2) + (- (SM / (sqrt 2))) is complex real ext-real set
(r `2) + (SM / (sqrt 2)) is complex real ext-real Element of REAL
].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[ is V195() V196() V197() open Element of bool REAL
(1,2,3,].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[,].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[,{0}) is Relation-like Function-like set
(1,2) --> (].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[,].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[) is Relation-like REAL -defined Function-like set
1 .--> ].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[ is Relation-like REAL -defined {1} -defined Function-like one-to-one constant trivial set
{1} --> ].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[ is Relation-like {1} -defined {].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[} -valued Function-like constant non empty total quasi_total Element of bool [:{1},{].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[}:]
{].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[} is non empty set
[:{1},{].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[}:] is Relation-like set
bool [:{1},{].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[}:] is set
2 .--> ].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[ is Relation-like REAL -defined {2} -defined Function-like one-to-one constant trivial set
{2} is non empty V195() V196() V197() V198() V199() V200() set
{2} --> ].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[ is Relation-like {2} -defined {].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[} -valued Function-like constant non empty total quasi_total Element of bool [:{2},{].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[}:]
{].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[} is non empty set
[:{2},{].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[}:] is Relation-like set
bool [:{2},{].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[}:] is set
(1 .--> ].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[) +* (2 .--> ].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[) is Relation-like REAL -defined Function-like set
3 .--> {0} is Relation-like REAL -defined {3} -defined Function-like one-to-one constant trivial set
{3} is non empty V195() V196() V197() V198() V199() V200() set
{3} --> {0} is Relation-like non-empty {3} -defined {{0}} -valued Function-like constant non empty total quasi_total Element of bool [:{3},{{0}}:]
{{0}} is non empty set
[:{3},{{0}}:] is Relation-like set
bool [:{3},{{0}}:] is set
((1,2) --> (].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[,].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[)) +* (3 .--> {0}) is Relation-like REAL -defined Function-like set
product (1,2,3,].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[,].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[,{0}) is set
p is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Euclid 3)
Ball (p,SM) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Euclid 3)
bool the carrier of (Euclid 3) is V74() set
x is set
dom (1,2,3,].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[,].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[,{0}) is set
y is Relation-like Function-like set
dom y is set
{ b₁ where b₁ is complex real ext-real Element of REAL : ( not b₁ <= (r `1) - (SM / (sqrt 2)) & not (r `1) + (SM / (sqrt 2)) <= b₁ ) } is set
{ b₁ where b₁ is complex real ext-real Element of REAL : ( not b₁ <= (r `2) - (SM / (sqrt 2)) & not (r `2) + (SM / (sqrt 2)) <= b₁ ) } is set
(1,2,3,].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[,].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[,{0}) . 1 is set
(1,2,3,].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[,].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[,{0}) . 2 is set
(1,2,3,].((r `1) - (SM / (sqrt 2))),((r `1) + (SM / (sqrt 2))).[,].((r `2) - (SM / (sqrt 2))),((r `2) + (SM / (sqrt 2))).[,{0}) . 3 is set
y . 1 is set
y . 2 is set
y . 3 is set
hx is complex real ext-real Element of REAL
hy is complex real ext-real Element of REAL
(r `1) - ((r `1) - (SM / (sqrt 2))) is complex real ext-real Element of REAL
- ((r `1) - (SM / (sqrt 2))) is complex real ext-real set
(r `1) + (- ((r `1) - (SM / (sqrt 2)))) is complex real ext-real set
(r `1) - ((r `1) + (SM / (sqrt 2))) is complex real ext-real Element of REAL
- ((r `1) + (SM / (sqrt 2))) is complex real ext-real set
(r `1) + (- ((r `1) + (SM / (sqrt 2)))) is complex real ext-real set
(SM / (sqrt 2)) + (SM / (sqrt 2)) is complex real ext-real Element of COMPLEX
- (SM / (sqrt 2)) is complex real ext-real Element of COMPLEX
(- (SM / (sqrt 2))) + (SM / (sqrt 2)) is complex real ext-real Element of COMPLEX
<*(y . 1),(y . 2),(y . 3)*> is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like set
<*(y . 1)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like set
[1,(y . 1)] is set
{1,(y . 1)} is non empty set
{{1,(y . 1)},{1}} is non empty set
{[1,(y . 1)]} is Relation-like Function-like non empty set
<*(y . 2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like set
[1,(y . 2)] is set
{1,(y . 2)} is non empty set
{{1,(y . 2)},{1}} is non empty set
{[1,(y . 2)]} is Relation-like Function-like non empty set
<*(y . 1)*> ^ <*(y . 2)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like set
<*(y . 3)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like set
[1,(y . 3)] is set
{1,(y . 3)} is non empty set
{{1,(y . 3)},{1}} is non empty set
{[1,(y . 3)]} is Relation-like Function-like non empty set
(<*(y . 1)*> ^ <*(y . 2)*>) ^ <*(y . 3)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like set
dom <*(y . 1),(y . 2),(y . 3)*> is non empty V195() V196() V197() V198() V199() V200() Element of bool NAT
len <*(y . 1),(y . 2),(y . 3)*> is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT
Seg (len <*(y . 1),(y . 2),(y . 3)*>) is V32() V39( len <*(y . 1),(y . 2),(y . 3)*>) V195() V196() V197() V198() V199() V200() Element of bool NAT
{ b₁ where b₁ is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT : ( 1 <= b₁ & b₁ <= len <*(y . 1),(y . 2),(y . 3)*> ) } is set
c₃₂ is set
y . c₃₂ is set
<*(y . 1),(y . 2),(y . 3)*> . c₃₂ is set
|[hx,hy,0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*hx*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,hx] is set
{1,hx} is non empty V195() V196() V197() set
{{1,hx},{1}} is non empty set
{[1,hx]} is Relation-like Function-like non empty set
<*hy*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,hy] is set
{1,hy} is non empty V195() V196() V197() set
{{1,hy},{1}} is non empty set
{[1,hy]} is Relation-like Function-like non empty set
<*hx*> ^ <*hy*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*hx*> ^ <*hy*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
c₃₂ is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
hx - (r `1) is complex real ext-real Element of REAL
- (r `1) is complex real ext-real set
hx + (- (r `1)) is complex real ext-real set
abs (hx - (r `1)) is complex real ext-real Element of REAL
Re (hx - (r `1)) is complex real ext-real Element of REAL
(Re (hx - (r `1))) ^2 is complex real ext-real non negative Element of REAL
(Re (hx - (r `1))) * (Re (hx - (r `1))) is complex real ext-real non negative set
Im (hx - (r `1)) is complex real ext-real Element of REAL
(Im (hx - (r `1))) ^2 is complex real ext-real non negative Element of REAL
(Im (hx - (r `1))) * (Im (hx - (r `1))) is complex real ext-real non negative set
((Re (hx - (r `1))) ^2) + ((Im (hx - (r `1))) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (hx - (r `1))) ^2) + ((Im (hx - (r `1))) ^2)) is complex real ext-real non negative Element of REAL
hy - (r `2) is complex real ext-real Element of REAL
- (r `2) is complex real ext-real set
hy + (- (r `2)) is complex real ext-real set
abs (hy - (r `2)) is complex real ext-real Element of REAL
Re (hy - (r `2)) is complex real ext-real Element of REAL
(Re (hy - (r `2))) ^2 is complex real ext-real non negative Element of REAL
(Re (hy - (r `2))) * (Re (hy - (r `2))) is complex real ext-real non negative set
Im (hy - (r `2)) is complex real ext-real Element of REAL
(Im (hy - (r `2))) ^2 is complex real ext-real non negative Element of REAL
(Im (hy - (r `2))) * (Im (hy - (r `2))) is complex real ext-real non negative set
((Re (hy - (r `2))) ^2) + ((Im (hy - (r `2))) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (hy - (r `2))) ^2) + ((Im (hy - (r `2))) ^2)) is complex real ext-real non negative Element of REAL
(SM / (sqrt 2)) ^2 is complex real ext-real non negative Element of COMPLEX
(SM / (sqrt 2)) * (SM / (sqrt 2)) is complex real ext-real non negative set
(abs (hx - (r `1))) ^2 is complex real ext-real non negative Element of REAL
(abs (hx - (r `1))) * (abs (hx - (r `1))) is complex real ext-real non negative set
(abs (hy - (r `2))) ^2 is complex real ext-real non negative Element of REAL
(abs (hy - (r `2))) * (abs (hy - (r `2))) is complex real ext-real non negative set
SM ^2 is complex real ext-real non negative set
SM * SM is complex real ext-real non negative set
(sqrt 2) ^2 is non empty complex real ext-real positive non negative Element of REAL
(sqrt 2) * (sqrt 2) is non empty complex real ext-real positive non negative set
(SM ^2) / ((sqrt 2) ^2) is complex real ext-real non negative Element of COMPLEX
((sqrt 2) ^2) " is non empty complex real ext-real positive non negative set
(SM ^2) * (((sqrt 2) ^2) ") is complex real ext-real non negative set
(SM ^2) / 2 is complex real ext-real non negative Element of COMPLEX
(SM ^2) * (2 ") is complex real ext-real non negative set
(hx - (r `1)) ^2 is complex real ext-real non negative Element of REAL
(hx - (r `1)) * (hx - (r `1)) is complex real ext-real non negative set
(hy - (r `2)) ^2 is complex real ext-real non negative Element of REAL
(hy - (r `2)) * (hy - (r `2)) is complex real ext-real non negative set
((SM ^2) / 2) + ((SM ^2) / 2) is complex real ext-real non negative Element of COMPLEX
((hx - (r `1)) ^2) + ((hy - (r `2)) ^2) is complex real ext-real non negative Element of REAL
sqrt (SM ^2) is complex real ext-real non negative set
sqrt (((hx - (r `1)) ^2) + ((hy - (r `2)) ^2)) is complex real ext-real non negative Element of REAL
c<sub>32</sub> `1 is complex real ext-real Element of REAL
c₃₂ `2 is complex real ext-real Element of REAL
Ch is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Euclid 3)
dist (Ch,p) is complex real ext-real Element of REAL
(Pitag_dist 3) . (Ch,p) is set
(c<sub>32</sub> `1) - (r `1) is complex real ext-real Element of REAL
(c<sub>32</sub> `1) + (- (r `1)) is complex real ext-real set
((c<sub>32</sub> `1) - (r `1)) ^2 is complex real ext-real non negative Element of REAL
((c<sub>32</sub> `1) - (r `1)) * ((c<sub>32</sub> `1) - (r `1)) is complex real ext-real non negative set
(c<sub>32</sub> `2) - (r `2) is complex real ext-real Element of REAL
(c<sub>32</sub> `2) + (- (r `2)) is complex real ext-real set
((c<sub>32</sub> `2) - (r `2)) ^2 is complex real ext-real non negative Element of REAL
((c<sub>32</sub> `2) - (r `2)) * ((c<sub>32</sub> `2) - (r `2)) is complex real ext-real non negative set
(((c<sub>32</sub> `1) - (r `1)) ^2) + (((c<sub>32</sub> `2) - (r `2)) ^2) is complex real ext-real non negative Element of REAL
c<sub>32</sub> `3 is complex real ext-real Element of REAL
(c₃₂ `3) - (r `3) is complex real ext-real Element of REAL
- (r `3) is complex real ext-real set
(c<sub>32</sub> `3) + (- (r `3)) is complex real ext-real set
((c<sub>32</sub> `3) - (r `3)) ^2 is complex real ext-real non negative Element of REAL
((c<sub>32</sub> `3) - (r `3)) * ((c<sub>32</sub> `3) - (r `3)) is complex real ext-real non negative set
((((c<sub>32</sub> `1) - (r `1)) ^2) + (((c<sub>32</sub> `2) - (r `2)) ^2)) + (((c<sub>32</sub> `3) - (r `3)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((((c<sub>32</sub> `1) - (r `1)) ^2) + (((c<sub>32</sub> `2) - (r `2)) ^2)) + (((c<sub>32</sub> `3) - (r `3)) ^2)) is complex real ext-real non negative Element of REAL
Q - Q is complex real integer ext-real set
- Q is complex real integer ext-real non positive set
Q + (- Q) is complex real integer ext-real set
(Q - Q) ^2 is complex real ext-real non negative set
(Q - Q) * (Q - Q) is complex real integer ext-real non negative set
((((c₃₂ `1) - (r `1)) ^2) + (((c₃₂ `2) - (r `2)) ^2)) + ((Q - Q) ^2) is complex real ext-real non negative Element of REAL
sqrt (((((c₃₂ `1) - (r `1)) ^2) + (((c₃₂ `2) - (r `2)) ^2)) + ((Q - Q) ^2)) is complex real ext-real non negative Element of REAL
((((c₃₂ `1) - (r `1)) ^2) + (((c₃₂ `2) - (r `2)) ^2)) + Q is complex real ext-real non negative Element of REAL
sqrt (((((c₃₂ `1) - (r `1)) ^2) + (((c₃₂ `2) - (r `2)) ^2)) + Q) is complex real ext-real non negative Element of REAL
SM is complex real integer ext-real set
(2 * PI) * SM is complex real ext-real Element of REAL
r is complex set
p is complex real ext-real Element of REAL
Rotate (r,p) is complex Element of COMPLEX
|.r.| is complex real ext-real Element of REAL
Re r is complex real ext-real Element of REAL
(Re r) ^2 is complex real ext-real non negative Element of REAL
(Re r) * (Re r) is complex real ext-real non negative set
Im r is complex real ext-real Element of REAL
(Im r) ^2 is complex real ext-real non negative Element of REAL
(Im r) * (Im r) is complex real ext-real non negative set
((Re r) ^2) + ((Im r) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re r) ^2) + ((Im r) ^2)) is complex real ext-real non negative Element of REAL
Arg r is complex real ext-real Element of REAL
p + (Arg r) is complex real ext-real Element of REAL
cos (p + (Arg r)) is complex real ext-real Element of REAL
|.r.| * (cos (p + (Arg r))) is complex real ext-real Element of REAL
sin (p + (Arg r)) is complex real ext-real Element of REAL
|.r.| * (sin (p + (Arg r))) is complex real ext-real Element of REAL
(|.r.| * (sin (p + (Arg r)))) * <i> is complex set
(|.r.| * (cos (p + (Arg r)))) + ((|.r.| * (sin (p + (Arg r)))) * <i>) is complex set
p + ((2 * PI) * SM) is complex real ext-real Element of REAL
Rotate (r,(p + ((2 * PI) * SM))) is complex Element of COMPLEX
(p + ((2 * PI) * SM)) + (Arg r) is complex real ext-real Element of REAL
cos ((p + ((2 * PI) * SM)) + (Arg r)) is complex real ext-real Element of REAL
|.r.| * (cos ((p + ((2 * PI) * SM)) + (Arg r))) is complex real ext-real Element of REAL
sin ((p + ((2 * PI) * SM)) + (Arg r)) is complex real ext-real Element of REAL
|.r.| * (sin ((p + ((2 * PI) * SM)) + (Arg r))) is complex real ext-real Element of REAL
(|.r.| * (sin ((p + ((2 * PI) * SM)) + (Arg r)))) * <i> is complex set
(|.r.| * (cos ((p + ((2 * PI) * SM)) + (Arg r)))) + ((|.r.| * (sin ((p + ((2 * PI) * SM)) + (Arg r)))) * <i>) is complex set
(p + (Arg r)) + ((2 * PI) * SM) is complex real ext-real Element of REAL
cos ((p + (Arg r)) + ((2 * PI) * SM)) is complex real ext-real Element of REAL
sin ((p + (Arg r)) + ((2 * PI) * SM)) is complex real ext-real Element of REAL
SM is complex real integer ext-real set
(2 * PI) * SM is complex real ext-real Element of REAL
r is complex real ext-real Element of REAL
Rotate r is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
r + ((2 * PI) * SM) is complex real ext-real Element of REAL
Rotate (r + ((2 * PI) * SM)) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
p is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate r) . p is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(Rotate (r + ((2 * PI) * SM))) . p is Relation-like Function-like complex-yielding ext-real-valued real-valued set
p `1 is complex real ext-real Element of REAL
K382(p,1) is complex real ext-real Element of REAL
p `2 is complex real ext-real Element of REAL
K382(p,2) is complex real ext-real Element of REAL
(p `2) * <i> is complex set
(p `1) + ((p `2) * <i>) is complex set
Rotate (((p `1) + ((p `2) * <i>)),r) is complex Element of COMPLEX
|.((p `1) + ((p `2) * <i>)).| is complex real ext-real Element of REAL
Re ((p `1) + ((p `2) * <i>)) is complex real ext-real Element of REAL
(Re ((p `1) + ((p `2) * <i>))) ^2 is complex real ext-real non negative Element of REAL
(Re ((p `1) + ((p `2) * <i>))) * (Re ((p `1) + ((p `2) * <i>))) is complex real ext-real non negative set
Im ((p `1) + ((p `2) * <i>)) is complex real ext-real Element of REAL
(Im ((p `1) + ((p `2) * <i>))) ^2 is complex real ext-real non negative Element of REAL
(Im ((p `1) + ((p `2) * <i>))) * (Im ((p `1) + ((p `2) * <i>))) is complex real ext-real non negative set
((Re ((p `1) + ((p `2) * <i>))) ^2) + ((Im ((p `1) + ((p `2) * <i>))) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re ((p `1) + ((p `2) * <i>))) ^2) + ((Im ((p `1) + ((p `2) * <i>))) ^2)) is complex real ext-real non negative Element of REAL
Arg ((p `1) + ((p `2) * <i>)) is complex real ext-real Element of REAL
r + (Arg ((p `1) + ((p `2) * <i>))) is complex real ext-real Element of REAL
cos (r + (Arg ((p `1) + ((p `2) * <i>)))) is complex real ext-real Element of REAL
|.((p `1) + ((p `2) * <i>)).| * (cos (r + (Arg ((p `1) + ((p `2) * <i>))))) is complex real ext-real Element of REAL
sin (r + (Arg ((p `1) + ((p `2) * <i>)))) is complex real ext-real Element of REAL
|.((p `1) + ((p `2) * <i>)).| * (sin (r + (Arg ((p `1) + ((p `2) * <i>))))) is complex real ext-real Element of REAL
(|.((p `1) + ((p `2) * <i>)).| * (sin (r + (Arg ((p `1) + ((p `2) * <i>)))))) * <i> is complex set
(|.((p `1) + ((p `2) * <i>)).| * (cos (r + (Arg ((p `1) + ((p `2) * <i>)))))) + ((|.((p `1) + ((p `2) * <i>)).| * (sin (r + (Arg ((p `1) + ((p `2) * <i>)))))) * <i>) is complex set
Rotate (((p `1) + ((p `2) * <i>)),(r + ((2 * PI) * SM))) is complex Element of COMPLEX
(r + ((2 * PI) * SM)) + (Arg ((p `1) + ((p `2) * <i>))) is complex real ext-real Element of REAL
cos ((r + ((2 * PI) * SM)) + (Arg ((p `1) + ((p `2) * <i>)))) is complex real ext-real Element of REAL
|.((p `1) + ((p `2) * <i>)).| * (cos ((r + ((2 * PI) * SM)) + (Arg ((p `1) + ((p `2) * <i>))))) is complex real ext-real Element of REAL
sin ((r + ((2 * PI) * SM)) + (Arg ((p `1) + ((p `2) * <i>)))) is complex real ext-real Element of REAL
|.((p `1) + ((p `2) * <i>)).| * (sin ((r + ((2 * PI) * SM)) + (Arg ((p `1) + ((p `2) * <i>))))) is complex real ext-real Element of REAL
(|.((p `1) + ((p `2) * <i>)).| * (sin ((r + ((2 * PI) * SM)) + (Arg ((p `1) + ((p `2) * <i>)))))) * <i> is complex set
(|.((p `1) + ((p `2) * <i>)).| * (cos ((r + ((2 * PI) * SM)) + (Arg ((p `1) + ((p `2) * <i>)))))) + ((|.((p `1) + ((p `2) * <i>)).| * (sin ((r + ((2 * PI) * SM)) + (Arg ((p `1) + ((p `2) * <i>)))))) * <i>) is complex set
(Rotate r) . p is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Re (Rotate (((p `1) + ((p `2) * <i>)),r)) is complex real ext-real Element of REAL
Im (Rotate (((p `1) + ((p `2) * <i>)),r)) is complex real ext-real Element of REAL
|[(Re (Rotate (((p `1) + ((p `2) * <i>)),r))),(Im (Rotate (((p `1) + ((p `2) * <i>)),r)))]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(Re (Rotate (((p `1) + ((p `2) * <i>)),r)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Re (Rotate (((p `1) + ((p `2) * <i>)),r)))] is set
{1,(Re (Rotate (((p `1) + ((p `2) * <i>)),r)))} is non empty V195() V196() V197() set
{{1,(Re (Rotate (((p `1) + ((p `2) * <i>)),r)))},{1}} is non empty set
{[1,(Re (Rotate (((p `1) + ((p `2) * <i>)),r)))]} is Relation-like Function-like non empty set
<*(Im (Rotate (((p `1) + ((p `2) * <i>)),r)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Im (Rotate (((p `1) + ((p `2) * <i>)),r)))] is set
{1,(Im (Rotate (((p `1) + ((p `2) * <i>)),r)))} is non empty V195() V196() V197() set
{{1,(Im (Rotate (((p `1) + ((p `2) * <i>)),r)))},{1}} is non empty set
{[1,(Im (Rotate (((p `1) + ((p `2) * <i>)),r)))]} is Relation-like Function-like non empty set
<*(Re (Rotate (((p `1) + ((p `2) * <i>)),r)))*> ^ <*(Im (Rotate (((p `1) + ((p `2) * <i>)),r)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(Rotate (r + ((2 * PI) * SM))) . p is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
SM is complex real ext-real Element of REAL
Rotate SM is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate SM) . r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
|.((Rotate SM) . r).| is complex real ext-real non negative Element of REAL
sqr ((Rotate SM) . r) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (((Rotate SM) . r),((Rotate SM) . r)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (((Rotate SM) . r),((Rotate SM) . r)) is Relation-like Function-like set
K254(((Rotate SM) . r),((Rotate SM) . r)) is Relation-like Function-like set
K254(((Rotate SM) . r),((Rotate SM) . r)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((Rotate SM) . r) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr ((Rotate SM) . r)) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((Rotate SM) . r))) is complex real ext-real Element of REAL
|.r.| is complex real ext-real non negative Element of REAL
sqr r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (r,r) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (r,r) is Relation-like Function-like set
K254(r,r) is Relation-like Function-like set
K254(r,r) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
r (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr r) is complex real ext-real Element of REAL
sqrt (Sum (sqr r)) is complex real ext-real Element of REAL
euc2cpx r is complex Element of COMPLEX
r `1 is complex real ext-real Element of REAL
K382(r,1) is complex real ext-real Element of REAL
r `2 is complex real ext-real Element of REAL
K382(r,2) is complex real ext-real Element of REAL
(r `2) * <i> is complex set
(r `1) + ((r `2) * <i>) is complex set
Rotate ((euc2cpx r),SM) is complex Element of COMPLEX
|.(euc2cpx r).| is complex real ext-real Element of REAL
Re (euc2cpx r) is complex real ext-real Element of REAL
(Re (euc2cpx r)) ^2 is complex real ext-real non negative Element of REAL
(Re (euc2cpx r)) * (Re (euc2cpx r)) is complex real ext-real non negative set
Im (euc2cpx r) is complex real ext-real Element of REAL
(Im (euc2cpx r)) ^2 is complex real ext-real non negative Element of REAL
(Im (euc2cpx r)) * (Im (euc2cpx r)) is complex real ext-real non negative set
((Re (euc2cpx r)) ^2) + ((Im (euc2cpx r)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (euc2cpx r)) ^2) + ((Im (euc2cpx r)) ^2)) is complex real ext-real non negative Element of REAL
Arg (euc2cpx r) is complex real ext-real Element of REAL
SM + (Arg (euc2cpx r)) is complex real ext-real Element of REAL
cos (SM + (Arg (euc2cpx r))) is complex real ext-real Element of REAL
|.(euc2cpx r).| * (cos (SM + (Arg (euc2cpx r)))) is complex real ext-real Element of REAL
sin (SM + (Arg (euc2cpx r))) is complex real ext-real Element of REAL
|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r)))) is complex real ext-real Element of REAL
(|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r))))) * <i> is complex set
(|.(euc2cpx r).| * (cos (SM + (Arg (euc2cpx r))))) + ((|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r))))) * <i>) is complex set
Re (Rotate ((euc2cpx r),SM)) is complex real ext-real Element of REAL
Im (Rotate ((euc2cpx r),SM)) is complex real ext-real Element of REAL
cpx2euc (Rotate ((euc2cpx r),SM)) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
|[(Re (Rotate ((euc2cpx r),SM))),(Im (Rotate ((euc2cpx r),SM)))]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(Re (Rotate ((euc2cpx r),SM)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Re (Rotate ((euc2cpx r),SM)))] is set
{1,(Re (Rotate ((euc2cpx r),SM)))} is non empty V195() V196() V197() set
{{1,(Re (Rotate ((euc2cpx r),SM)))},{1}} is non empty set
{[1,(Re (Rotate ((euc2cpx r),SM)))]} is Relation-like Function-like non empty set
<*(Im (Rotate ((euc2cpx r),SM)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Im (Rotate ((euc2cpx r),SM)))] is set
{1,(Im (Rotate ((euc2cpx r),SM)))} is non empty V195() V196() V197() set
{{1,(Im (Rotate ((euc2cpx r),SM)))},{1}} is non empty set
{[1,(Im (Rotate ((euc2cpx r),SM)))]} is Relation-like Function-like non empty set
<*(Re (Rotate ((euc2cpx r),SM)))*> ^ <*(Im (Rotate ((euc2cpx r),SM)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((Rotate SM) . r) `1 is complex real ext-real Element of REAL
K382(((Rotate SM) . r),1) is complex real ext-real Element of REAL
((Rotate SM) . r) `2 is complex real ext-real Element of REAL
K382(((Rotate SM) . r),2) is complex real ext-real Element of REAL
|.r.| ^2 is complex real ext-real non negative Element of REAL
|.r.| * |.r.| is complex real ext-real non negative set
|.(euc2cpx r).| ^2 is complex real ext-real non negative Element of REAL
|.(euc2cpx r).| * |.(euc2cpx r).| is complex real ext-real non negative set
(|.(euc2cpx r).| ^2) * 1 is complex real ext-real non negative Element of REAL
(cos (SM + (Arg (euc2cpx r)))) ^2 is complex real ext-real non negative Element of REAL
(cos (SM + (Arg (euc2cpx r)))) * (cos (SM + (Arg (euc2cpx r)))) is complex real ext-real non negative set
(sin (SM + (Arg (euc2cpx r)))) ^2 is complex real ext-real non negative Element of REAL
(sin (SM + (Arg (euc2cpx r)))) * (sin (SM + (Arg (euc2cpx r)))) is complex real ext-real non negative set
((cos (SM + (Arg (euc2cpx r)))) ^2) + ((sin (SM + (Arg (euc2cpx r)))) ^2) is complex real ext-real non negative Element of REAL
(|.(euc2cpx r).| ^2) * (((cos (SM + (Arg (euc2cpx r)))) ^2) + ((sin (SM + (Arg (euc2cpx r)))) ^2)) is complex real ext-real non negative Element of REAL
(|.(euc2cpx r).| * (cos (SM + (Arg (euc2cpx r))))) ^2 is complex real ext-real non negative Element of REAL
(|.(euc2cpx r).| * (cos (SM + (Arg (euc2cpx r))))) * (|.(euc2cpx r).| * (cos (SM + (Arg (euc2cpx r))))) is complex real ext-real non negative set
(|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r))))) ^2 is complex real ext-real non negative Element of REAL
(|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r))))) * (|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r))))) is complex real ext-real non negative set
((|.(euc2cpx r).| * (cos (SM + (Arg (euc2cpx r))))) ^2) + ((|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r))))) ^2) is complex real ext-real non negative Element of REAL
|.((Rotate SM) . r).| ^2 is complex real ext-real non negative Element of REAL
|.((Rotate SM) . r).| * |.((Rotate SM) . r).| is complex real ext-real non negative set
SM is complex real ext-real Element of REAL
Rotate SM is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate SM) . r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Arg ((Rotate SM) . r) is complex real ext-real Element of REAL
euc2cpx ((Rotate SM) . r) is complex Element of COMPLEX
((Rotate SM) . r) `1 is complex real ext-real Element of REAL
K382(((Rotate SM) . r),1) is complex real ext-real Element of REAL
((Rotate SM) . r) `2 is complex real ext-real Element of REAL
K382(((Rotate SM) . r),2) is complex real ext-real Element of REAL
(((Rotate SM) . r) `2) * <i> is complex set
(((Rotate SM) . r) `1) + ((((Rotate SM) . r) `2) * <i>) is complex set
Arg (euc2cpx ((Rotate SM) . r)) is complex real ext-real Element of REAL
euc2cpx r is complex Element of COMPLEX
r `1 is complex real ext-real Element of REAL
K382(r,1) is complex real ext-real Element of REAL
r `2 is complex real ext-real Element of REAL
K382(r,2) is complex real ext-real Element of REAL
(r `2) * <i> is complex set
(r `1) + ((r `2) * <i>) is complex set
Rotate ((euc2cpx r),SM) is complex Element of COMPLEX
|.(euc2cpx r).| is complex real ext-real Element of REAL
Re (euc2cpx r) is complex real ext-real Element of REAL
(Re (euc2cpx r)) ^2 is complex real ext-real non negative Element of REAL
(Re (euc2cpx r)) * (Re (euc2cpx r)) is complex real ext-real non negative set
Im (euc2cpx r) is complex real ext-real Element of REAL
(Im (euc2cpx r)) ^2 is complex real ext-real non negative Element of REAL
(Im (euc2cpx r)) * (Im (euc2cpx r)) is complex real ext-real non negative set
((Re (euc2cpx r)) ^2) + ((Im (euc2cpx r)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (euc2cpx r)) ^2) + ((Im (euc2cpx r)) ^2)) is complex real ext-real non negative Element of REAL
Arg (euc2cpx r) is complex real ext-real Element of REAL
SM + (Arg (euc2cpx r)) is complex real ext-real Element of REAL
cos (SM + (Arg (euc2cpx r))) is complex real ext-real Element of REAL
|.(euc2cpx r).| * (cos (SM + (Arg (euc2cpx r)))) is complex real ext-real Element of REAL
sin (SM + (Arg (euc2cpx r))) is complex real ext-real Element of REAL
|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r)))) is complex real ext-real Element of REAL
(|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r))))) * <i> is complex set
(|.(euc2cpx r).| * (cos (SM + (Arg (euc2cpx r))))) + ((|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r))))) * <i>) is complex set
Arg (Rotate ((euc2cpx r),SM)) is complex real ext-real Element of REAL
cpx2euc (Rotate ((euc2cpx r),SM)) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Re (Rotate ((euc2cpx r),SM)) is complex real ext-real Element of REAL
Im (Rotate ((euc2cpx r),SM)) is complex real ext-real Element of REAL
|[(Re (Rotate ((euc2cpx r),SM))),(Im (Rotate ((euc2cpx r),SM)))]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(Re (Rotate ((euc2cpx r),SM)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Re (Rotate ((euc2cpx r),SM)))] is set
{1,(Re (Rotate ((euc2cpx r),SM)))} is non empty V195() V196() V197() set
{{1,(Re (Rotate ((euc2cpx r),SM)))},{1}} is non empty set
{[1,(Re (Rotate ((euc2cpx r),SM)))]} is Relation-like Function-like non empty set
<*(Im (Rotate ((euc2cpx r),SM)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Im (Rotate ((euc2cpx r),SM)))] is set
{1,(Im (Rotate ((euc2cpx r),SM)))} is non empty V195() V196() V197() set
{{1,(Im (Rotate ((euc2cpx r),SM)))},{1}} is non empty set
{[1,(Im (Rotate ((euc2cpx r),SM)))]} is Relation-like Function-like non empty set
<*(Re (Rotate ((euc2cpx r),SM)))*> ^ <*(Im (Rotate ((euc2cpx r),SM)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
SM is complex real ext-real Element of REAL
Rotate SM is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate SM) . r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Arg ((Rotate SM) . r) is complex real ext-real Element of REAL
euc2cpx ((Rotate SM) . r) is complex Element of COMPLEX
((Rotate SM) . r) `1 is complex real ext-real Element of REAL
K382(((Rotate SM) . r),1) is complex real ext-real Element of REAL
((Rotate SM) . r) `2 is complex real ext-real Element of REAL
K382(((Rotate SM) . r),2) is complex real ext-real Element of REAL
(((Rotate SM) . r) `2) * <i> is complex set
(((Rotate SM) . r) `1) + ((((Rotate SM) . r) `2) * <i>) is complex set
Arg (euc2cpx ((Rotate SM) . r)) is complex real ext-real Element of REAL
Arg r is complex real ext-real Element of REAL
euc2cpx r is complex Element of COMPLEX
r `1 is complex real ext-real Element of REAL
K382(r,1) is complex real ext-real Element of REAL
r `2 is complex real ext-real Element of REAL
K382(r,2) is complex real ext-real Element of REAL
(r `2) * <i> is complex set
(r `1) + ((r `2) * <i>) is complex set
Arg (euc2cpx r) is complex real ext-real Element of REAL
SM + (Arg r) is complex real ext-real Element of REAL
Rotate ((euc2cpx r),SM) is complex Element of COMPLEX
|.(euc2cpx r).| is complex real ext-real Element of REAL
Re (euc2cpx r) is complex real ext-real Element of REAL
(Re (euc2cpx r)) ^2 is complex real ext-real non negative Element of REAL
(Re (euc2cpx r)) * (Re (euc2cpx r)) is complex real ext-real non negative set
Im (euc2cpx r) is complex real ext-real Element of REAL
(Im (euc2cpx r)) ^2 is complex real ext-real non negative Element of REAL
(Im (euc2cpx r)) * (Im (euc2cpx r)) is complex real ext-real non negative set
((Re (euc2cpx r)) ^2) + ((Im (euc2cpx r)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (euc2cpx r)) ^2) + ((Im (euc2cpx r)) ^2)) is complex real ext-real non negative Element of REAL
SM + (Arg (euc2cpx r)) is complex real ext-real Element of REAL
cos (SM + (Arg (euc2cpx r))) is complex real ext-real Element of REAL
|.(euc2cpx r).| * (cos (SM + (Arg (euc2cpx r)))) is complex real ext-real Element of REAL
sin (SM + (Arg (euc2cpx r))) is complex real ext-real Element of REAL
|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r)))) is complex real ext-real Element of REAL
(|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r))))) * <i> is complex set
(|.(euc2cpx r).| * (cos (SM + (Arg (euc2cpx r))))) + ((|.(euc2cpx r).| * (sin (SM + (Arg (euc2cpx r))))) * <i>) is complex set
Arg (Rotate ((euc2cpx r),SM)) is complex real ext-real Element of REAL
f is complex real integer ext-real set
(2 * PI) * f is complex real ext-real Element of REAL
((2 * PI) * f) + (SM + (Arg (euc2cpx r))) is complex real ext-real Element of REAL
SM is complex real ext-real Element of REAL
Rotate SM is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
(Rotate SM) . (0. (TOP-REAL 2)) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Rotate ((euc2cpx (0. (TOP-REAL 2))),SM) is complex Element of COMPLEX
|.(euc2cpx (0. (TOP-REAL 2))).| is complex real ext-real Element of REAL
Re (euc2cpx (0. (TOP-REAL 2))) is complex real ext-real Element of REAL
(Re (euc2cpx (0. (TOP-REAL 2)))) ^2 is complex real ext-real non negative Element of REAL
(Re (euc2cpx (0. (TOP-REAL 2)))) * (Re (euc2cpx (0. (TOP-REAL 2)))) is complex real ext-real non negative set
Im (euc2cpx (0. (TOP-REAL 2))) is complex real ext-real Element of REAL
(Im (euc2cpx (0. (TOP-REAL 2)))) ^2 is complex real ext-real non negative Element of REAL
(Im (euc2cpx (0. (TOP-REAL 2)))) * (Im (euc2cpx (0. (TOP-REAL 2)))) is complex real ext-real non negative set
((Re (euc2cpx (0. (TOP-REAL 2)))) ^2) + ((Im (euc2cpx (0. (TOP-REAL 2)))) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (euc2cpx (0. (TOP-REAL 2)))) ^2) + ((Im (euc2cpx (0. (TOP-REAL 2)))) ^2)) is complex real ext-real non negative Element of REAL
Arg (euc2cpx (0. (TOP-REAL 2))) is complex real ext-real Element of REAL
SM + (Arg (euc2cpx (0. (TOP-REAL 2)))) is complex real ext-real Element of REAL
cos (SM + (Arg (euc2cpx (0. (TOP-REAL 2))))) is complex real ext-real Element of REAL
|.(euc2cpx (0. (TOP-REAL 2))).| * (cos (SM + (Arg (euc2cpx (0. (TOP-REAL 2)))))) is complex real ext-real Element of REAL
sin (SM + (Arg (euc2cpx (0. (TOP-REAL 2))))) is complex real ext-real Element of REAL
|.(euc2cpx (0. (TOP-REAL 2))).| * (sin (SM + (Arg (euc2cpx (0. (TOP-REAL 2)))))) is complex real ext-real Element of REAL
(|.(euc2cpx (0. (TOP-REAL 2))).| * (sin (SM + (Arg (euc2cpx (0. (TOP-REAL 2))))))) * <i> is complex set
(|.(euc2cpx (0. (TOP-REAL 2))).| * (cos (SM + (Arg (euc2cpx (0. (TOP-REAL 2))))))) + ((|.(euc2cpx (0. (TOP-REAL 2))).| * (sin (SM + (Arg (euc2cpx (0. (TOP-REAL 2))))))) * <i>) is complex set
cpx2euc (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Re (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)) is complex real ext-real Element of REAL
Im (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)) is complex real ext-real Element of REAL
|[(Re (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM))),(Im (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(Re (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Re (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))] is set
{1,(Re (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))} is non empty V195() V196() V197() set
{{1,(Re (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))},{1}} is non empty set
{[1,(Re (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))]} is Relation-like Function-like non empty set
<*(Im (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(Im (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))] is set
{1,(Im (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))} is non empty V195() V196() V197() set
{{1,(Im (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))},{1}} is non empty set
{[1,(Im (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))]} is Relation-like Function-like non empty set
<*(Re (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))*> ^ <*(Im (Rotate ((euc2cpx (0. (TOP-REAL 2))),SM)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
SM is complex real ext-real Element of REAL
Rotate SM is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate SM) . r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
|.r.| is complex real ext-real non negative Element of REAL
sqr r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (r,r) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (r,r) is Relation-like Function-like set
K254(r,r) is Relation-like Function-like set
K254(r,r) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
r (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr r) is complex real ext-real Element of REAL
sqrt (Sum (sqr r)) is complex real ext-real Element of REAL
|.((Rotate SM) . r).| is complex real ext-real non negative Element of REAL
sqr ((Rotate SM) . r) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (((Rotate SM) . r),((Rotate SM) . r)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (((Rotate SM) . r),((Rotate SM) . r)) is Relation-like Function-like set
K254(((Rotate SM) . r),((Rotate SM) . r)) is Relation-like Function-like set
K254(((Rotate SM) . r),((Rotate SM) . r)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((Rotate SM) . r) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr ((Rotate SM) . r)) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((Rotate SM) . r))) is complex real ext-real Element of REAL
SM is complex real ext-real Element of REAL
Rotate SM is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
- SM is complex real ext-real Element of REAL
Rotate (- SM) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate (- SM)) . r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate SM) . ((Rotate (- SM)) . r) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Arg ((Rotate (- SM)) . r) is complex real ext-real Element of REAL
euc2cpx ((Rotate (- SM)) . r) is complex Element of COMPLEX
((Rotate (- SM)) . r) `1 is complex real ext-real Element of REAL
K382(((Rotate (- SM)) . r),1) is complex real ext-real Element of REAL
((Rotate (- SM)) . r) `2 is complex real ext-real Element of REAL
K382(((Rotate (- SM)) . r),2) is complex real ext-real Element of REAL
(((Rotate (- SM)) . r) `2) * <i> is complex set
(((Rotate (- SM)) . r) `1) + ((((Rotate (- SM)) . r) `2) * <i>) is complex set
Arg (euc2cpx ((Rotate (- SM)) . r)) is complex real ext-real Element of REAL
Arg r is complex real ext-real Element of REAL
euc2cpx r is complex Element of COMPLEX
r `1 is complex real ext-real Element of REAL
K382(r,1) is complex real ext-real Element of REAL
r `2 is complex real ext-real Element of REAL
K382(r,2) is complex real ext-real Element of REAL
(r `2) * <i> is complex set
(r `1) + ((r `2) * <i>) is complex set
Arg (euc2cpx r) is complex real ext-real Element of REAL
(- SM) + (Arg r) is complex real ext-real Element of REAL
e is complex real integer ext-real set
(2 * PI) * e is complex real ext-real Element of REAL
((- SM) + (Arg r)) + ((2 * PI) * e) is complex real ext-real Element of REAL
Arg ((Rotate SM) . ((Rotate (- SM)) . r)) is complex real ext-real Element of REAL
euc2cpx ((Rotate SM) . ((Rotate (- SM)) . r)) is complex Element of COMPLEX
((Rotate SM) . ((Rotate (- SM)) . r)) `1 is complex real ext-real Element of REAL
K382(((Rotate SM) . ((Rotate (- SM)) . r)),1) is complex real ext-real Element of REAL
((Rotate SM) . ((Rotate (- SM)) . r)) `2 is complex real ext-real Element of REAL
K382(((Rotate SM) . ((Rotate (- SM)) . r)),2) is complex real ext-real Element of REAL
(((Rotate SM) . ((Rotate (- SM)) . r)) `2) * <i> is complex set
(((Rotate SM) . ((Rotate (- SM)) . r)) `1) + ((((Rotate SM) . ((Rotate (- SM)) . r)) `2) * <i>) is complex set
Arg (euc2cpx ((Rotate SM) . ((Rotate (- SM)) . r))) is complex real ext-real Element of REAL
SM + (Arg ((Rotate (- SM)) . r)) is complex real ext-real Element of REAL
f1 is complex real integer ext-real set
(2 * PI) * f1 is complex real ext-real Element of REAL
(SM + (Arg ((Rotate (- SM)) . r))) + ((2 * PI) * f1) is complex real ext-real Element of REAL
|.((Rotate SM) . ((Rotate (- SM)) . r)).| is complex real ext-real non negative Element of REAL
sqr ((Rotate SM) . ((Rotate (- SM)) . r)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (((Rotate SM) . ((Rotate (- SM)) . r)),((Rotate SM) . ((Rotate (- SM)) . r))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (((Rotate SM) . ((Rotate (- SM)) . r)),((Rotate SM) . ((Rotate (- SM)) . r))) is Relation-like Function-like set
K254(((Rotate SM) . ((Rotate (- SM)) . r)),((Rotate SM) . ((Rotate (- SM)) . r))) is Relation-like Function-like set
K254(((Rotate SM) . ((Rotate (- SM)) . r)),((Rotate SM) . ((Rotate (- SM)) . r))) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((Rotate SM) . ((Rotate (- SM)) . r)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr ((Rotate SM) . ((Rotate (- SM)) . r))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((Rotate SM) . ((Rotate (- SM)) . r)))) is complex real ext-real Element of REAL
|.((Rotate (- SM)) . r).| is complex real ext-real non negative Element of REAL
sqr ((Rotate (- SM)) . r) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (((Rotate (- SM)) . r),((Rotate (- SM)) . r)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (((Rotate (- SM)) . r),((Rotate (- SM)) . r)) is Relation-like Function-like set
K254(((Rotate (- SM)) . r),((Rotate (- SM)) . r)) is Relation-like Function-like set
K254(((Rotate (- SM)) . r),((Rotate (- SM)) . r)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((Rotate (- SM)) . r) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr ((Rotate (- SM)) . r)) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((Rotate (- SM)) . r))) is complex real ext-real Element of REAL
|.r.| is complex real ext-real non negative Element of REAL
sqr r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (r,r) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (r,r) is Relation-like Function-like set
K254(r,r) is Relation-like Function-like set
K254(r,r) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
r (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr r) is complex real ext-real Element of REAL
sqrt (Sum (sqr r)) is complex real ext-real Element of REAL
e + f1 is complex real integer ext-real set
(2 * PI) * (e + f1) is complex real ext-real Element of REAL
(Arg r) + ((2 * PI) * (e + f1)) is complex real ext-real Element of REAL
(Rotate (- SM)) . (0. (TOP-REAL 2)) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate SM) . ((Rotate (- SM)) . (0. (TOP-REAL 2))) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate SM) . (0. (TOP-REAL 2)) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
id (TOP-REAL 2) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
id the carrier of (TOP-REAL 2) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like one-to-one non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
SM is complex real ext-real Element of REAL
- SM is complex real ext-real Element of REAL
Rotate (- SM) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
Rotate SM is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
(Rotate SM) * (Rotate (- SM)) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
((Rotate SM) * (Rotate (- SM))) . r is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(id (TOP-REAL 2)) . r is Relation-like Function-like complex-yielding ext-real-valued real-valued set
((Rotate SM) * (Rotate (- SM))) . r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate (- SM)) . r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate SM) . ((Rotate (- SM)) . r) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(id (TOP-REAL 2)) . r is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
SM is complex real ext-real set
Sphere ((0. (TOP-REAL 2)),SM) is functional closed V214( TOP-REAL 2) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 2)
r is complex real ext-real Element of REAL
Rotate r is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
p is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate r) . p is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
|.p.| is complex real ext-real non negative Element of REAL
sqr p is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (p,p) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (p,p) is Relation-like Function-like set
K254(p,p) is Relation-like Function-like set
K254(p,p) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
p (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr p) is complex real ext-real Element of REAL
sqrt (Sum (sqr p)) is complex real ext-real Element of REAL
|.((Rotate r) . p).| is complex real ext-real non negative Element of REAL
sqr ((Rotate r) . p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (((Rotate r) . p),((Rotate r) . p)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (((Rotate r) . p),((Rotate r) . p)) is Relation-like Function-like set
K254(((Rotate r) . p),((Rotate r) . p)) is Relation-like Function-like set
K254(((Rotate r) . p),((Rotate r) . p)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((Rotate r) . p) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr ((Rotate r) . p)) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((Rotate r) . p))) is complex real ext-real Element of REAL
((Rotate r) . p) - (0. (TOP-REAL 2)) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
((Rotate r) . p) - (0. (TOP-REAL 2)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (0. (TOP-REAL 2)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (0. (TOP-REAL 2)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(0. (TOP-REAL 2)) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(0. (TOP-REAL 2)) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((Rotate r) . p) + (- (0. (TOP-REAL 2))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: (((Rotate r) . p),(- (0. (TOP-REAL 2)))) is Relation-like Function-like set
diffreal .: (((Rotate r) . p),(0. (TOP-REAL 2))) is Relation-like Function-like set
K254(((Rotate r) . p),(0. (TOP-REAL 2))) is Relation-like Function-like set
K254(((Rotate r) . p),(0. (TOP-REAL 2))) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
- r is complex real ext-real Element of REAL
Rotate (- r) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
- (- r) is complex real ext-real Element of REAL
Rotate (- (- r)) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
(Rotate (- (- r))) . p is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate (- r)) . ((Rotate (- (- r))) . p) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate (- r)) . ((Rotate r) . p) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
((Rotate (- r)) . ((Rotate r) . p)) - (0. (TOP-REAL 2)) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
((Rotate (- r)) . ((Rotate r) . p)) - (0. (TOP-REAL 2)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
((Rotate (- r)) . ((Rotate r) . p)) + (- (0. (TOP-REAL 2))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: (((Rotate (- r)) . ((Rotate r) . p)),(- (0. (TOP-REAL 2)))) is Relation-like Function-like set
diffreal .: (((Rotate (- r)) . ((Rotate r) . p)),(0. (TOP-REAL 2))) is Relation-like Function-like set
K254(((Rotate (- r)) . ((Rotate r) . p)),(0. (TOP-REAL 2))) is Relation-like Function-like set
K254(((Rotate (- r)) . ((Rotate r) . p)),(0. (TOP-REAL 2))) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
SM is complex real ext-real non negative set
Sphere ((0. (TOP-REAL 2)),SM) is functional non empty closed V214( TOP-REAL 2) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 2)
r is complex real ext-real Element of REAL
Rotate r is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
(Rotate r) .: (Sphere ((0. (TOP-REAL 2)),SM)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 2)
e is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
f1 is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate r) . f1 is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
e is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
- r is complex real ext-real Element of REAL
Rotate (- r) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
(Rotate (- r)) . e is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(Rotate r) . ((Rotate (- r)) . e) is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
r is complex real ext-real Element of REAL
Rotate r is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
SM is complex real ext-real non negative set
Tcircle ((0. (TOP-REAL 2)),SM) is non empty strict TopSpace-like pathwise_connected complex-functions-membered real-functions-membered connected SubSpace of TOP-REAL 2
Sphere ((0. (TOP-REAL 2)),SM) is functional non empty closed V214( TOP-REAL 2) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 2)
(TOP-REAL 2) | (Sphere ((0. (TOP-REAL 2)),SM)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 2
(Rotate r) | (Tcircle ((0. (TOP-REAL 2)),SM)) is Relation-like the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (TOP-REAL 2):]
the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (TOP-REAL 2):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (TOP-REAL 2):] is set
(Rotate r) | the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) -defined the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
[: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)):] is set
dom ((Rotate r) | (Tcircle ((0. (TOP-REAL 2)),SM))) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL 2)),SM))
bool the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) is V74() set
f1 is set
((Rotate r) | (Tcircle ((0. (TOP-REAL 2)),SM))) . f1 is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(Rotate r) . f1 is Relation-like Function-like complex-yielding ext-real-valued real-valued set
SM is complex real ext-real non negative set
Tcircle ((0. (TOP-REAL 2)),SM) is non empty strict TopSpace-like pathwise_connected complex-functions-membered real-functions-membered connected SubSpace of TOP-REAL 2
Sphere ((0. (TOP-REAL 2)),SM) is functional non empty closed V214( TOP-REAL 2) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 2)
(TOP-REAL 2) | (Sphere ((0. (TOP-REAL 2)),SM)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 2
r is complex real ext-real Element of REAL
(SM,r) is Relation-like the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) -defined the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)):]
the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)):] is set
Rotate r is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
(Rotate r) | (Tcircle ((0. (TOP-REAL 2)),SM)) is Relation-like the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (TOP-REAL 2):]
[: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (TOP-REAL 2):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)), the carrier of (TOP-REAL 2):] is set
(Rotate r) | the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) -defined the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
[#] (Tcircle ((0. (TOP-REAL 2)),SM)) is functional non empty non proper open closed dense non boundary complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL 2)),SM))
bool the carrier of (Tcircle ((0. (TOP-REAL 2)),SM)) is V74() set
[#] (TOP-REAL 2) is functional non empty non proper open closed dense non boundary complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 2)
the topology of (TOP-REAL 2) is non empty Element of bool (bool the carrier of (TOP-REAL 2))
bool (bool the carrier of (TOP-REAL 2)) is set
TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) is non empty strict TopSpace-like TopStruct
f1 is Relation-like the carrier of (TopSpaceMetr (Euclid 2)) -defined the carrier of (TopSpaceMetr (Euclid 2)) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (TopSpaceMetr (Euclid 2)), the carrier of (TopSpaceMetr (Euclid 2)):]
bool the carrier of (TopSpaceMetr (Euclid 2)) is V74() set
e is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 2)
(TOP-REAL 2) | e is strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 2
h is Element of bool the carrier of (TopSpaceMetr (Euclid 2))
(TopSpaceMetr (Euclid 2)) | h is strict TopSpace-like SubSpace of TopSpaceMetr (Euclid 2)
(Rotate r) .: e is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 2)
SM is complex real ext-real set
CircleMap . SM is Relation-like Function-like complex-yielding ext-real-valued real-valued set
r is complex real ext-real set
CircleMap . r is Relation-like Function-like complex-yielding ext-real-valued real-valued set
r - SM is complex real ext-real set
- SM is complex real ext-real set
r + (- SM) is complex real ext-real set
CircleMap . (r - SM) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
p is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Arg p is complex real ext-real Element of REAL
euc2cpx p is complex Element of COMPLEX
p `1 is complex real ext-real Element of REAL
K382(p,1) is complex real ext-real Element of REAL
p `2 is complex real ext-real Element of REAL
K382(p,2) is complex real ext-real Element of REAL
(p `2) * <i> is complex set
(p `1) + ((p `2) * <i>) is complex set
Arg (euc2cpx p) is complex real ext-real Element of REAL
- (Arg p) is complex real ext-real Element of REAL
(1,(- (Arg p))) is Relation-like the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) -defined the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) -valued Function-like one-to-one non empty total quasi_total onto bijective continuous Function-yielding V178() being_homeomorphism complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (Tcircle ((0. (TOP-REAL 2)),1)):]
the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (Tcircle ((0. (TOP-REAL 2)),1)):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (Tcircle ((0. (TOP-REAL 2)),1)):] is set
Rotate (- (Arg p)) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
(Rotate (- (Arg p))) | (Tcircle ((0. (TOP-REAL 2)),1)) is Relation-like the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (TOP-REAL 2):]
[: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (TOP-REAL 2):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (TOP-REAL 2):] is set
(Rotate (- (Arg p))) | the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) -defined the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
(1,(- (Arg p))) . (CircleMap . r) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
R^1 r is complex real ext-real Element of the carrier of K708()
CircleMap . (R^1 r) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
R^1 (r - SM) is complex real ext-real Element of the carrier of K708()
CircleMap . (R^1 (r - SM)) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
e is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
|.e.| is complex real ext-real non negative Element of REAL
sqr e is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (e,e) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (e,e) is Relation-like Function-like set
K254(e,e) is Relation-like Function-like set
K254(e,e) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
e (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr e) is complex real ext-real Element of REAL
sqrt (Sum (sqr e)) is complex real ext-real Element of REAL
(Rotate (- (Arg p))) . e is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Arg ((Rotate (- (Arg p))) . e) is complex real ext-real Element of REAL
euc2cpx ((Rotate (- (Arg p))) . e) is complex Element of COMPLEX
((Rotate (- (Arg p))) . e) `1 is complex real ext-real Element of REAL
K382(((Rotate (- (Arg p))) . e),1) is complex real ext-real Element of REAL
((Rotate (- (Arg p))) . e) `2 is complex real ext-real Element of REAL
K382(((Rotate (- (Arg p))) . e),2) is complex real ext-real Element of REAL
(((Rotate (- (Arg p))) . e) `2) * <i> is complex set
(((Rotate (- (Arg p))) . e) `1) + ((((Rotate (- (Arg p))) . e) `2) * <i>) is complex set
Arg (euc2cpx ((Rotate (- (Arg p))) . e)) is complex real ext-real Element of REAL
Arg e is complex real ext-real Element of REAL
euc2cpx e is complex Element of COMPLEX
e `1 is complex real ext-real Element of REAL
K382(e,1) is complex real ext-real Element of REAL
e `2 is complex real ext-real Element of REAL
K382(e,2) is complex real ext-real Element of REAL
(e `2) * <i> is complex set
(e `1) + ((e `2) * <i>) is complex set
Arg (euc2cpx e) is complex real ext-real Element of REAL
(- (Arg p)) + (Arg e) is complex real ext-real Element of REAL
h is complex real integer ext-real set
(2 * PI) * h is complex real ext-real Element of REAL
((- (Arg p)) + (Arg e)) + ((2 * PI) * h) is complex real ext-real Element of REAL
frac SM is complex real ext-real Element of REAL
[\SM/] is complex real integer ext-real set
SM - [\SM/] is complex real ext-real set
- [\SM/] is complex real integer ext-real set
SM + (- [\SM/]) is complex real ext-real set
(2 * PI) * (frac SM) is complex real ext-real Element of REAL
|.((Rotate (- (Arg p))) . e).| is complex real ext-real non negative Element of REAL
sqr ((Rotate (- (Arg p))) . e) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (((Rotate (- (Arg p))) . e),((Rotate (- (Arg p))) . e)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (((Rotate (- (Arg p))) . e),((Rotate (- (Arg p))) . e)) is Relation-like Function-like set
K254(((Rotate (- (Arg p))) . e),((Rotate (- (Arg p))) . e)) is Relation-like Function-like set
K254(((Rotate (- (Arg p))) . e),((Rotate (- (Arg p))) . e)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((Rotate (- (Arg p))) . e) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr ((Rotate (- (Arg p))) . e)) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((Rotate (- (Arg p))) . e))) is complex real ext-real Element of REAL
f1 is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
|.f1.| is complex real ext-real non negative Element of REAL
sqr f1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (f1,f1) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (f1,f1) is Relation-like Function-like set
K254(f1,f1) is Relation-like Function-like set
K254(f1,f1) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
f1 (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr f1) is complex real ext-real Element of REAL
sqrt (Sum (sqr f1)) is complex real ext-real Element of REAL
frac (r - SM) is complex real ext-real Element of REAL
[\(r - SM)/] is complex real integer ext-real set
(r - SM) - [\(r - SM)/] is complex real ext-real set
- [\(r - SM)/] is complex real integer ext-real set
(r - SM) + (- [\(r - SM)/]) is complex real ext-real set
frac r is complex real ext-real Element of REAL
[\r/] is complex real integer ext-real set
r - [\r/] is complex real ext-real set
- [\r/] is complex real integer ext-real set
r + (- [\r/]) is complex real ext-real set
(frac r) - (frac SM) is complex real ext-real Element of REAL
- (frac SM) is complex real ext-real set
(frac r) + (- (frac SM)) is complex real ext-real set
x is complex real integer ext-real set
((frac r) - (frac SM)) + x is complex real ext-real Element of REAL
- ((2 * PI) * (frac SM)) is complex real ext-real Element of REAL
(2 * PI) * (frac r) is complex real ext-real Element of REAL
(- ((2 * PI) * (frac SM))) + ((2 * PI) * (frac r)) is complex real ext-real Element of REAL
((- ((2 * PI) * (frac SM))) + ((2 * PI) * (frac r))) + ((2 * PI) * h) is complex real ext-real Element of REAL
(2 * PI) * (frac (r - SM)) is complex real ext-real Element of REAL
h - x is complex real integer ext-real set
- x is complex real integer ext-real set
h + (- x) is complex real integer ext-real set
(2 * PI) * (h - x) is complex real ext-real Element of REAL
((2 * PI) * (frac (r - SM))) + ((2 * PI) * (h - x)) is complex real ext-real Element of REAL
Arg f1 is complex real ext-real Element of REAL
euc2cpx f1 is complex Element of COMPLEX
f1 `1 is complex real ext-real Element of REAL
K382(f1,1) is complex real ext-real Element of REAL
f1 `2 is complex real ext-real Element of REAL
K382(f1,2) is complex real ext-real Element of REAL
(f1 `2) * <i> is complex set
(f1 `1) + ((f1 `2) * <i>) is complex set
Arg (euc2cpx f1) is complex real ext-real Element of REAL
(Arg f1) + ((2 * PI) * (h - x)) is complex real ext-real Element of REAL
SM is non empty complex ordinal natural real integer ext-real positive non negative set
TOP-REAL SM is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL SM) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
Tunit_circle SM is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL SM
0. (TOP-REAL SM) is Relation-like NAT -defined Function-like V32() V39(SM) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
the ZeroF of (TOP-REAL SM) is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
Tcircle ((0. (TOP-REAL SM)),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL SM
Sphere ((0. (TOP-REAL SM)),1) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL SM)
bool the carrier of (TOP-REAL SM) is V74() set
(TOP-REAL SM) | (Sphere ((0. (TOP-REAL SM)),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL SM
the carrier of (Tunit_circle SM) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p is complex real ext-real non negative set
Tcircle (r,p) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL SM
Sphere (r,p) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL SM)
(TOP-REAL SM) | (Sphere (r,p)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL SM
the carrier of (Tcircle (r,p)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tunit_circle SM), the carrier of (Tcircle (r,p)):] is Relation-like set
bool [: the carrier of (Tunit_circle SM), the carrier of (Tcircle (r,p)):] is set
f1 is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle SM)
h is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p * h is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p * h is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
p multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] (p,(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> p is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{p} is non empty V195() V196() V197() set
[:(dom (id REAL)),{p}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> p),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> p),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
h (#) (p multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p * h) + r is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
(p * h) + r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((p * h),r) is Relation-like Function-like set
K254((p * h),r) is Relation-like Function-like set
K254((p * h),r) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((p * h) + r) - r is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
((p * h) + r) - r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- r is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * r is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
r (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
r (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((p * h) + r) + (- r) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: (((p * h) + r),(- r)) is Relation-like Function-like set
diffreal .: (((p * h) + r),r) is Relation-like Function-like set
K254(((p * h) + r),r) is Relation-like Function-like set
K254(((p * h) + r),r) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|.(((p * h) + r) - r).| is complex real ext-real non negative Element of REAL
sqr (((p * h) + r) - r) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((((p * h) + r) - r),(((p * h) + r) - r)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((((p * h) + r) - r),(((p * h) + r) - r)) is Relation-like Function-like set
K254((((p * h) + r) - r),(((p * h) + r) - r)) is Relation-like Function-like set
K254((((p * h) + r) - r),(((p * h) + r) - r)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(((p * h) + r) - r) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (((p * h) + r) - r)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (((p * h) + r) - r))) is complex real ext-real Element of REAL
|.(p * h).| is complex real ext-real non negative Element of REAL
sqr (p * h) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((p * h),(p * h)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((p * h),(p * h)) is Relation-like Function-like set
K254((p * h),(p * h)) is Relation-like Function-like set
K254((p * h),(p * h)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(p * h) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (p * h)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (p * h))) is complex real ext-real Element of REAL
abs p is complex real ext-real Element of REAL
Re p is complex real ext-real Element of REAL
(Re p) ^2 is complex real ext-real non negative Element of REAL
(Re p) * (Re p) is complex real ext-real non negative set
Im p is complex real ext-real Element of REAL
(Im p) ^2 is complex real ext-real non negative Element of REAL
(Im p) * (Im p) is complex real ext-real non negative set
((Re p) ^2) + ((Im p) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re p) ^2) + ((Im p) ^2)) is complex real ext-real non negative Element of REAL
|.h.| is complex real ext-real non negative Element of REAL
sqr h is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (h,h) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (h,h) is Relation-like Function-like set
K254(h,h) is Relation-like Function-like set
K254(h,h) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
h (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr h) is complex real ext-real Element of REAL
sqrt (Sum (sqr h)) is complex real ext-real Element of REAL
(abs p) * |.h.| is complex real ext-real Element of REAL
p * |.h.| is complex real ext-real non negative Element of REAL
p * 1 is complex real ext-real non negative Element of REAL
y is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle (r,p))
f1 is Relation-like the carrier of (Tunit_circle SM) -defined the carrier of (Tcircle (r,p)) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle SM), the carrier of (Tcircle (r,p)):]
h is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle SM)
f1 . h is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle (r,p))
x is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p * x is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p * x is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
p multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] (p,(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> p is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{p} is non empty V195() V196() V197() set
[:(dom (id REAL)),{p}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> p),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> p),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
x (#) (p multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p * x) + r is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
(p * x) + r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((p * x),r) is Relation-like Function-like set
K254((p * x),r) is Relation-like Function-like set
K254((p * x),r) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
y is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p * y is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p * y is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
y (#) (p multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p * y) + r is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
(p * y) + r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((p * y),r) is Relation-like Function-like set
K254((p * y),r) is Relation-like Function-like set
K254((p * y),r) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
f is Relation-like the carrier of (Tunit_circle SM) -defined the carrier of (Tcircle (r,p)) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle SM), the carrier of (Tcircle (r,p)):]
e is Relation-like the carrier of (Tunit_circle SM) -defined the carrier of (Tcircle (r,p)) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle SM), the carrier of (Tcircle (r,p)):]
f1 is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle SM)
f . f1 is Relation-like Function-like complex-yielding ext-real-valued real-valued set
e . f1 is Relation-like Function-like complex-yielding ext-real-valued real-valued set
f . f1 is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle (r,p))
h is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p * h is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p * h is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
p multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] (p,(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> p is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{p} is non empty V195() V196() V197() set
[:(dom (id REAL)),{p}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> p),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> p),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
h (#) (p multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p * h) + r is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
(p * h) + r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((p * h),r) is Relation-like Function-like set
K254((p * h),r) is Relation-like Function-like set
K254((p * h),r) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
e . f1 is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle (r,p))
SM is non empty complex ordinal natural real integer ext-real positive non negative set
TOP-REAL SM is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL SM) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
Tunit_circle SM is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL SM
0. (TOP-REAL SM) is Relation-like NAT -defined Function-like V32() V39(SM) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
the ZeroF of (TOP-REAL SM) is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
Tcircle ((0. (TOP-REAL SM)),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL SM
Sphere ((0. (TOP-REAL SM)),1) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL SM)
bool the carrier of (TOP-REAL SM) is V74() set
(TOP-REAL SM) | (Sphere ((0. (TOP-REAL SM)),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL SM
r is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p is non empty complex real ext-real positive non negative set
Tcircle (r,p) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL SM
Sphere (r,p) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL SM)
(TOP-REAL SM) | (Sphere (r,p)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL SM
(SM,r,p) is Relation-like the carrier of (Tunit_circle SM) -defined the carrier of (Tcircle (r,p)) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle SM), the carrier of (Tcircle (r,p)):]
the carrier of (Tunit_circle SM) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
the carrier of (Tcircle (r,p)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tunit_circle SM), the carrier of (Tcircle (r,p)):] is Relation-like set
bool [: the carrier of (Tunit_circle SM), the carrier of (Tcircle (r,p)):] is set
f is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle SM)
e is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
(SM,r,p) . f is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle (r,p))
p * e is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
p * e is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
p multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] (p,(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> p is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{p} is non empty V195() V196() V197() set
[:(dom (id REAL)),{p}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> p),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> p),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
e (#) (p multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p * e) + r is Relation-like NAT -defined Function-like V32() V39(SM) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL SM)
(p * e) + r is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((p * e),r) is Relation-like Function-like set
K254((p * e),r) is Relation-like Function-like set
K254((p * e),r) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
[: the carrier of R^1, the carrier of (Tunit_circle 3):] is Relation-like set
bool [: the carrier of R^1, the carrier of (Tunit_circle 3):] is set
SM is complex real ext-real Element of the carrier of R^1
(2 * PI) * SM is complex real ext-real Element of REAL
cos ((2 * PI) * SM) is complex real ext-real Element of REAL
sin ((2 * PI) * SM) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos ((2 * PI) * SM))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos ((2 * PI) * SM))] is set
{1,(cos ((2 * PI) * SM))} is non empty V195() V196() V197() set
{{1,(cos ((2 * PI) * SM))},{1}} is non empty set
{[1,(cos ((2 * PI) * SM))]} is Relation-like Function-like non empty set
<*(sin ((2 * PI) * SM))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin ((2 * PI) * SM))] is set
{1,(sin ((2 * PI) * SM))} is non empty V195() V196() V197() set
{{1,(sin ((2 * PI) * SM))},{1}} is non empty set
{[1,(sin ((2 * PI) * SM))]} is Relation-like Function-like non empty set
<*(cos ((2 * PI) * SM))*> ^ <*(sin ((2 * PI) * SM))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos ((2 * PI) * SM))*> ^ <*(sin ((2 * PI) * SM))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]| is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]| is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- |[0,0,0]| is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * |[0,0,0]| is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|[0,0,0]| (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[0,0,0]| (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| + (- |[0,0,0]|) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: (|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|,(- |[0,0,0]|)) is Relation-like Function-like set
diffreal .: (|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|,|[0,0,0]|) is Relation-like Function-like set
K254(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|,|[0,0,0]|) is Relation-like Function-like set
K254(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|,|[0,0,0]|) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|.(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|).| is complex real ext-real non negative Element of REAL
sqr (|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|),(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|),(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|)) is Relation-like Function-like set
K254((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|),(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|)) is Relation-like Function-like set
K254((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|),(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| - |[0,0,0]|))) is complex real ext-real Element of REAL
|.|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|.| is complex real ext-real non negative Element of REAL
sqr |[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|,|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|,|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|) is Relation-like Function-like set
K254(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|,|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|) is Relation-like Function-like set
K254(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|,|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr |[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|) is complex real ext-real Element of REAL
sqrt (Sum (sqr |[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]|)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `1 is complex real ext-real Element of REAL
(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `1) ^2 is complex real ext-real non negative Element of REAL
(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `1) * (|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `1) is complex real ext-real non negative set
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `2 is complex real ext-real Element of REAL
(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `2) ^2 is complex real ext-real non negative Element of REAL
(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `2) * (|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `2) is complex real ext-real non negative set
((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `1) ^2) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `2) ^2) is complex real ext-real non negative Element of REAL
|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `3 is complex real ext-real Element of REAL
(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `3) ^2 is complex real ext-real non negative Element of REAL
(|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `3) * (|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `3) is complex real ext-real non negative set
(((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `1) ^2) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `2) ^2)) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `3) ^2) is complex real ext-real non negative Element of REAL
sqrt ((((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `1) ^2) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `2) ^2)) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `3) ^2)) is complex real ext-real non negative Element of REAL
(cos ((2 * PI) * SM)) ^2 is complex real ext-real non negative Element of REAL
(cos ((2 * PI) * SM)) * (cos ((2 * PI) * SM)) is complex real ext-real non negative set
((cos ((2 * PI) * SM)) ^2) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `2) ^2) is complex real ext-real non negative Element of REAL
(((cos ((2 * PI) * SM)) ^2) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `2) ^2)) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `3) ^2) is complex real ext-real non negative Element of REAL
sqrt ((((cos ((2 * PI) * SM)) ^2) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `2) ^2)) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `3) ^2)) is complex real ext-real non negative Element of REAL
(sin ((2 * PI) * SM)) ^2 is complex real ext-real non negative Element of REAL
(sin ((2 * PI) * SM)) * (sin ((2 * PI) * SM)) is complex real ext-real non negative set
((cos ((2 * PI) * SM)) ^2) + ((sin ((2 * PI) * SM)) ^2) is complex real ext-real non negative Element of REAL
(((cos ((2 * PI) * SM)) ^2) + ((sin ((2 * PI) * SM)) ^2)) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `3) ^2) is complex real ext-real non negative Element of REAL
sqrt ((((cos ((2 * PI) * SM)) ^2) + ((sin ((2 * PI) * SM)) ^2)) + ((|[(cos ((2 * PI) * SM)),(sin ((2 * PI) * SM)),0]| `3) ^2)) is complex real ext-real non negative Element of REAL
Q ^2 is complex real ext-real non negative set
Q * Q is complex ordinal natural real integer ext-real non negative set
(((cos ((2 * PI) * SM)) ^2) + ((sin ((2 * PI) * SM)) ^2)) + (Q ^2) is complex real ext-real non negative Element of REAL
sqrt ((((cos ((2 * PI) * SM)) ^2) + ((sin ((2 * PI) * SM)) ^2)) + (Q ^2)) is complex real ext-real non negative Element of REAL
SM is Relation-like the carrier of R^1 -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of R^1, the carrier of (Tunit_circle 3):]
r is complex real ext-real set
SM . r is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(2 * PI) * r is complex real ext-real Element of REAL
cos ((2 * PI) * r) is complex real ext-real Element of REAL
sin ((2 * PI) * r) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * r)),(sin ((2 * PI) * r)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos ((2 * PI) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos ((2 * PI) * r))] is set
{1,(cos ((2 * PI) * r))} is non empty V195() V196() V197() set
{{1,(cos ((2 * PI) * r))},{1}} is non empty set
{[1,(cos ((2 * PI) * r))]} is Relation-like Function-like non empty set
<*(sin ((2 * PI) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin ((2 * PI) * r))] is set
{1,(sin ((2 * PI) * r))} is non empty V195() V196() V197() set
{{1,(sin ((2 * PI) * r))},{1}} is non empty set
{[1,(sin ((2 * PI) * r))]} is Relation-like Function-like non empty set
<*(cos ((2 * PI) * r))*> ^ <*(sin ((2 * PI) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos ((2 * PI) * r))*> ^ <*(sin ((2 * PI) * r))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
SM is Relation-like the carrier of R^1 -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of R^1, the carrier of (Tunit_circle 3):]
r is Relation-like the carrier of R^1 -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of R^1, the carrier of (Tunit_circle 3):]
p is complex real ext-real Element of the carrier of R^1
SM . p is Relation-like Function-like complex-yielding ext-real-valued real-valued set
r . p is Relation-like Function-like complex-yielding ext-real-valued real-valued set
SM . p is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
(2 * PI) * p is complex real ext-real Element of REAL
cos ((2 * PI) * p) is complex real ext-real Element of REAL
sin ((2 * PI) * p) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * p)),(sin ((2 * PI) * p)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos ((2 * PI) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos ((2 * PI) * p))] is set
{1,(cos ((2 * PI) * p))} is non empty V195() V196() V197() set
{{1,(cos ((2 * PI) * p))},{1}} is non empty set
{[1,(cos ((2 * PI) * p))]} is Relation-like Function-like non empty set
<*(sin ((2 * PI) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin ((2 * PI) * p))] is set
{1,(sin ((2 * PI) * p))} is non empty V195() V196() V197() set
{{1,(sin ((2 * PI) * p))},{1}} is non empty set
{[1,(sin ((2 * PI) * p))]} is Relation-like Function-like non empty set
<*(cos ((2 * PI) * p))*> ^ <*(sin ((2 * PI) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos ((2 * PI) * p))*> ^ <*(sin ((2 * PI) * p))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
r . p is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
() is Relation-like the carrier of R^1 -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of R^1, the carrier of (Tunit_circle 3):]
r is complex real integer ext-real set
() . r is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(2 * PI) * r is complex real ext-real Element of REAL
((2 * PI) * r) + Q is complex real ext-real Element of REAL
cos (((2 * PI) * r) + Q) is complex real ext-real Element of REAL
sin ((2 * PI) * r) is complex real ext-real Element of REAL
(sin ((2 * PI) * r)) + Q is complex real ext-real Element of REAL
|[(cos (((2 * PI) * r) + Q)),((sin ((2 * PI) * r)) + Q),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos (((2 * PI) * r) + Q))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos (((2 * PI) * r) + Q))] is set
{1,(cos (((2 * PI) * r) + Q))} is non empty V195() V196() V197() set
{{1,(cos (((2 * PI) * r) + Q))},{1}} is non empty set
{[1,(cos (((2 * PI) * r) + Q))]} is Relation-like Function-like non empty set
<*((sin ((2 * PI) * r)) + Q)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((sin ((2 * PI) * r)) + Q)] is set
{1,((sin ((2 * PI) * r)) + Q)} is non empty V195() V196() V197() set
{{1,((sin ((2 * PI) * r)) + Q)},{1}} is non empty set
{[1,((sin ((2 * PI) * r)) + Q)]} is Relation-like Function-like non empty set
<*(cos (((2 * PI) * r) + Q))*> ^ <*((sin ((2 * PI) * r)) + Q)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos (((2 * PI) * r) + Q))*> ^ <*((sin ((2 * PI) * r)) + Q)*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
sin (((2 * PI) * r) + Q) is complex real ext-real Element of REAL
|[(cos 0),(sin (((2 * PI) * r) + Q)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos 0)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos 0)] is set
{1,(cos 0)} is non empty V195() V196() V197() set
{{1,(cos 0)},{1}} is non empty set
{[1,(cos 0)]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * r) + Q))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * r) + Q))] is set
{1,(sin (((2 * PI) * r) + Q))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * r) + Q))},{1}} is non empty set
{[1,(sin (((2 * PI) * r) + Q))]} is Relation-like Function-like non empty set
<*(cos 0)*> ^ <*(sin (((2 * PI) * r) + Q))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos 0)*> ^ <*(sin (((2 * PI) * r) + Q))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
R^1 (dom sin) is non empty V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (dom sin)) is non empty strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (dom sin))) is non empty V195() V196() V197() set
rng sin is non empty V195() V196() V197() Element of bool REAL
R^1 (rng sin) is non empty V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (rng sin)) is non empty strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (rng sin))) is non empty V195() V196() V197() set
R^1 sin is Relation-like the carrier of (K708() | (R^1 (dom sin))) -defined the carrier of (K708() | (R^1 (rng sin))) -valued Function-like non empty total quasi_total onto continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (K708() | (R^1 (dom sin))), the carrier of (K708() | (R^1 (rng sin))):]
[: the carrier of (K708() | (R^1 (dom sin))), the carrier of (K708() | (R^1 (rng sin))):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (K708() | (R^1 (dom sin))), the carrier of (K708() | (R^1 (rng sin))):] is set
R^1 | (R^1 (dom sin)) is non empty strict TopSpace-like V231() SubSpace of R^1
R^1 (dom cos) is non empty V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (dom cos)) is non empty strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (dom cos))) is non empty V195() V196() V197() set
rng cos is non empty V195() V196() V197() Element of bool REAL
R^1 (rng cos) is non empty V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (rng cos)) is non empty strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (rng cos))) is non empty V195() V196() V197() set
R^1 cos is Relation-like the carrier of (K708() | (R^1 (dom cos))) -defined the carrier of (K708() | (R^1 (rng cos))) -valued Function-like non empty total quasi_total onto continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (K708() | (R^1 (dom cos))), the carrier of (K708() | (R^1 (rng cos))):]
[: the carrier of (K708() | (R^1 (dom cos))), the carrier of (K708() | (R^1 (rng cos))):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (K708() | (R^1 (dom cos))), the carrier of (K708() | (R^1 (rng cos))):] is set
R^1 | (R^1 (dom cos)) is non empty strict TopSpace-like V231() SubSpace of R^1
AffineMap ((2 * PI),Q) is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total quasi_total onto bijective complex-yielding ext-real-valued real-valued continuous Element of bool [:REAL,REAL:]
cos * (AffineMap ((2 * PI),Q)) is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued continuous Element of bool [:REAL,REAL:]
AffineMap ((2 * PI),0) is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total quasi_total onto bijective complex-yielding ext-real-valued real-valued continuous Element of bool [:REAL,REAL:]
sin * (AffineMap ((2 * PI),0)) is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued continuous Element of bool [:REAL,REAL:]
r is complex real ext-real set
() . r is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(cos * (AffineMap ((2 * PI),Q))) . r is complex real ext-real set
(sin * (AffineMap ((2 * PI),0))) . r is complex real ext-real set
|[((cos * (AffineMap ((2 * PI),Q))) . r),((sin * (AffineMap ((2 * PI),0))) . r),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*((cos * (AffineMap ((2 * PI),Q))) . r)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((cos * (AffineMap ((2 * PI),Q))) . r)] is set
{1,((cos * (AffineMap ((2 * PI),Q))) . r)} is non empty V195() V196() V197() set
{{1,((cos * (AffineMap ((2 * PI),Q))) . r)},{1}} is non empty set
{[1,((cos * (AffineMap ((2 * PI),Q))) . r)]} is Relation-like Function-like non empty set
<*((sin * (AffineMap ((2 * PI),0))) . r)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((sin * (AffineMap ((2 * PI),0))) . r)] is set
{1,((sin * (AffineMap ((2 * PI),0))) . r)} is non empty V195() V196() V197() set
{{1,((sin * (AffineMap ((2 * PI),0))) . r)},{1}} is non empty set
{[1,((sin * (AffineMap ((2 * PI),0))) . r)]} is Relation-like Function-like non empty set
<*((cos * (AffineMap ((2 * PI),Q))) . r)*> ^ <*((sin * (AffineMap ((2 * PI),0))) . r)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*((cos * (AffineMap ((2 * PI),Q))) . r)*> ^ <*((sin * (AffineMap ((2 * PI),0))) . r)*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(2 * PI) * r is complex real ext-real Element of REAL
((2 * PI) * r) + Q is complex real ext-real Element of REAL
cos (((2 * PI) * r) + Q) is complex real ext-real Element of REAL
sin ((2 * PI) * r) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * r) + Q)),(sin ((2 * PI) * r)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos (((2 * PI) * r) + Q))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos (((2 * PI) * r) + Q))] is set
{1,(cos (((2 * PI) * r) + Q))} is non empty V195() V196() V197() set
{{1,(cos (((2 * PI) * r) + Q))},{1}} is non empty set
{[1,(cos (((2 * PI) * r) + Q))]} is Relation-like Function-like non empty set
<*(sin ((2 * PI) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin ((2 * PI) * r))] is set
{1,(sin ((2 * PI) * r))} is non empty V195() V196() V197() set
{{1,(sin ((2 * PI) * r))},{1}} is non empty set
{[1,(sin ((2 * PI) * r))]} is Relation-like Function-like non empty set
<*(cos (((2 * PI) * r) + Q))*> ^ <*(sin ((2 * PI) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos (((2 * PI) * r) + Q))*> ^ <*(sin ((2 * PI) * r))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
cos * (AffineMap ((2 * PI),0)) is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued continuous Element of bool [:REAL,REAL:]
(cos * (AffineMap ((2 * PI),0))) . r is complex real ext-real set
sin (((2 * PI) * r) + Q) is complex real ext-real Element of REAL
|[((cos * (AffineMap ((2 * PI),0))) . r),(sin (((2 * PI) * r) + Q)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*((cos * (AffineMap ((2 * PI),0))) . r)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((cos * (AffineMap ((2 * PI),0))) . r)] is set
{1,((cos * (AffineMap ((2 * PI),0))) . r)} is non empty V195() V196() V197() set
{{1,((cos * (AffineMap ((2 * PI),0))) . r)},{1}} is non empty set
{[1,((cos * (AffineMap ((2 * PI),0))) . r)]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * r) + Q))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * r) + Q))] is set
{1,(sin (((2 * PI) * r) + Q))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * r) + Q))},{1}} is non empty set
{[1,(sin (((2 * PI) * r) + Q))]} is Relation-like Function-like non empty set
<*((cos * (AffineMap ((2 * PI),0))) . r)*> ^ <*(sin (((2 * PI) * r) + Q))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*((cos * (AffineMap ((2 * PI),0))) . r)*> ^ <*(sin (((2 * PI) * r) + Q))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[((cos * (AffineMap ((2 * PI),0))) . r),((sin * (AffineMap ((2 * PI),0))) . r),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*((cos * (AffineMap ((2 * PI),0))) . r)*> ^ <*((sin * (AffineMap ((2 * PI),0))) . r)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*((cos * (AffineMap ((2 * PI),0))) . r)*> ^ <*((sin * (AffineMap ((2 * PI),0))) . r)*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
dom (AffineMap ((2 * PI),0)) is non empty V195() V196() V197() Element of bool REAL
R^1 (dom (AffineMap ((2 * PI),0))) is non empty V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (dom (AffineMap ((2 * PI),0)))) is non empty strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (dom (AffineMap ((2 * PI),0))))) is non empty V195() V196() V197() set
rng (AffineMap ((2 * PI),0)) is non empty V195() V196() V197() Element of bool REAL
R^1 (rng (AffineMap ((2 * PI),0))) is non empty V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (rng (AffineMap ((2 * PI),0)))) is non empty strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (rng (AffineMap ((2 * PI),0))))) is non empty V195() V196() V197() set
R^1 (AffineMap ((2 * PI),0)) is Relation-like the carrier of (K708() | (R^1 (dom (AffineMap ((2 * PI),0))))) -defined the carrier of (K708() | (R^1 (rng (AffineMap ((2 * PI),0))))) -valued Function-like one-to-one non empty total quasi_total onto bijective continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (K708() | (R^1 (dom (AffineMap ((2 * PI),0))))), the carrier of (K708() | (R^1 (rng (AffineMap ((2 * PI),0))))):]
[: the carrier of (K708() | (R^1 (dom (AffineMap ((2 * PI),0))))), the carrier of (K708() | (R^1 (rng (AffineMap ((2 * PI),0))))):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (K708() | (R^1 (dom (AffineMap ((2 * PI),0))))), the carrier of (K708() | (R^1 (rng (AffineMap ((2 * PI),0))))):] is set
R^1 (dom sin) is non empty V195() V196() V197() Element of bool the carrier of K708()
R^1 | (R^1 (dom sin)) is non empty strict TopSpace-like V231() SubSpace of R^1
R^1 (dom cos) is non empty V195() V196() V197() Element of bool the carrier of K708()
R^1 | (R^1 (dom cos)) is non empty strict TopSpace-like V231() SubSpace of R^1
R^1 sin is Relation-like the carrier of (K708() | (R^1 (dom sin))) -defined the carrier of (K708() | (R^1 (rng sin))) -valued Function-like non empty total quasi_total onto continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (K708() | (R^1 (dom sin))), the carrier of (K708() | (R^1 (rng sin))):]
K708() | (R^1 (dom sin)) is non empty strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (dom sin))) is non empty V195() V196() V197() set
rng sin is non empty V195() V196() V197() Element of bool REAL
R^1 (rng sin) is non empty V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (rng sin)) is non empty strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (rng sin))) is non empty V195() V196() V197() set
[: the carrier of (K708() | (R^1 (dom sin))), the carrier of (K708() | (R^1 (rng sin))):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (K708() | (R^1 (dom sin))), the carrier of (K708() | (R^1 (rng sin))):] is set
R^1 cos is Relation-like the carrier of (K708() | (R^1 (dom cos))) -defined the carrier of (K708() | (R^1 (rng cos))) -valued Function-like non empty total quasi_total onto continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (K708() | (R^1 (dom cos))), the carrier of (K708() | (R^1 (rng cos))):]
K708() | (R^1 (dom cos)) is non empty strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (dom cos))) is non empty V195() V196() V197() set
rng cos is non empty V195() V196() V197() Element of bool REAL
R^1 (rng cos) is non empty V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (rng cos)) is non empty strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (rng cos))) is non empty V195() V196() V197() set
[: the carrier of (K708() | (R^1 (dom cos))), the carrier of (K708() | (R^1 (rng cos))):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (K708() | (R^1 (dom cos))), the carrier of (K708() | (R^1 (rng cos))):] is set
f1 is Relation-like the carrier of R^1 -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of R^1, the carrier of R^1:]
f is Relation-like the carrier of R^1 -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of R^1, the carrier of R^1:]
f * f1 is Relation-like the carrier of R^1 -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of R^1, the carrier of R^1:]
e is Relation-like the carrier of R^1 -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of R^1, the carrier of R^1:]
e * f1 is Relation-like the carrier of R^1 -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of R^1, the carrier of R^1:]
[: the carrier of R^1, the carrier of (TOP-REAL 3):] is Relation-like set
bool [: the carrier of R^1, the carrier of (TOP-REAL 3):] is set
y is Relation-like the carrier of R^1 -defined the carrier of (TOP-REAL 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of R^1, the carrier of (TOP-REAL 3):]
hx is complex real ext-real Element of the carrier of R^1
y . hx is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
hy is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 3)
Int hy is functional open complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 3)
(e * f1) . hx is complex real ext-real Element of the carrier of R^1
(f * f1) . hx is complex real ext-real Element of the carrier of R^1
c<sub>32</sub> is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Euclid 3)
Ch is complex real ext-real set
Ball (c<sub>32</sub>,Ch) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Euclid 3)
bool the carrier of (Euclid 3) is V74() set
(y . hx) `1 is complex real ext-real Element of REAL
Ch / (sqrt 2) is complex real ext-real Element of COMPLEX
(sqrt 2) " is non empty complex real ext-real positive non negative set
Ch * ((sqrt 2) ") is complex real ext-real set
((y . hx) `1) - (Ch / (sqrt 2)) is complex real ext-real Element of REAL
- (Ch / (sqrt 2)) is complex real ext-real set
((y . hx) `1) + (- (Ch / (sqrt 2))) is complex real ext-real set
((y . hx) `1) + (Ch / (sqrt 2)) is complex real ext-real Element of REAL
].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[ is V195() V196() V197() open Element of bool REAL
(y . hx) `2 is complex real ext-real Element of REAL
((y . hx) `2) - (Ch / (sqrt 2)) is complex real ext-real Element of REAL
((y . hx) `2) + (- (Ch / (sqrt 2))) is complex real ext-real set
((y . hx) `2) + (Ch / (sqrt 2)) is complex real ext-real Element of REAL
].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[ is V195() V196() V197() open Element of bool REAL
(1,2,3,].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[,].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[,{0}) is Relation-like Function-like set
(1,2) --> (].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[,].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[) is Relation-like REAL -defined Function-like set
1 .--> ].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[ is Relation-like REAL -defined {1} -defined Function-like one-to-one constant trivial set
{1} --> ].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[ is Relation-like {1} -defined {].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[} -valued Function-like constant non empty total quasi_total Element of bool [:{1},{].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[}:]
{].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[} is non empty set
[:{1},{].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[}:] is Relation-like set
bool [:{1},{].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[}:] is set
2 .--> ].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[ is Relation-like REAL -defined {2} -defined Function-like one-to-one constant trivial set
{2} is non empty V195() V196() V197() V198() V199() V200() set
{2} --> ].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[ is Relation-like {2} -defined {].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[} -valued Function-like constant non empty total quasi_total Element of bool [:{2},{].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[}:]
{].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[} is non empty set
[:{2},{].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[}:] is Relation-like set
bool [:{2},{].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[}:] is set
(1 .--> ].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[) +* (2 .--> ].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[) is Relation-like REAL -defined Function-like set
3 .--> {0} is Relation-like REAL -defined {3} -defined Function-like one-to-one constant trivial set
{3} is non empty V195() V196() V197() V198() V199() V200() set
{3} --> {0} is Relation-like non-empty {3} -defined {{0}} -valued Function-like constant non empty total quasi_total Element of bool [:{3},{{0}}:]
{{0}} is non empty set
[:{3},{{0}}:] is Relation-like set
bool [:{3},{{0}}:] is set
((1,2) --> (].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[,].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[)) +* (3 .--> {0}) is Relation-like REAL -defined Function-like set
|[((e * f1) . hx),((f * f1) . hx),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*((e * f1) . hx)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((e * f1) . hx)] is set
{1,((e * f1) . hx)} is non empty V195() V196() V197() set
{{1,((e * f1) . hx)},{1}} is non empty set
{[1,((e * f1) . hx)]} is Relation-like Function-like non empty set
<*((f * f1) . hx)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((f * f1) . hx)] is set
{1,((f * f1) . hx)} is non empty V195() V196() V197() set
{{1,((f * f1) . hx)},{1}} is non empty set
{[1,((f * f1) . hx)]} is Relation-like Function-like non empty set
<*((e * f1) . hx)*> ^ <*((f * f1) . hx)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*((e * f1) . hx)*> ^ <*((f * f1) . hx)*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(y . hx) `3 is complex real ext-real Element of REAL
product (1,2,3,].(((y . hx) `1) - (Ch / (sqrt 2))),(((y . hx) `1) + (Ch / (sqrt 2))).[,].(((y . hx) `2) - (Ch / (sqrt 2))),(((y . hx) `2) + (Ch / (sqrt 2))).[,{0}) is set
F is V195() V196() V197() Element of bool the carrier of R^1
S is V195() V196() V197() Element of bool the carrier of R^1
A is V195() V196() V197() Element of bool the carrier of R^1
(f * f1) .: A is V195() V196() V197() Element of bool the carrier of R^1
H is V195() V196() V197() Element of bool the carrier of R^1
(e * f1) .: H is V195() V196() V197() Element of bool the carrier of R^1
A /\ H is V195() V196() V197() Element of bool the carrier of R^1
y .: (A /\ H) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 3)
t is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
E is complex real ext-real Element of the carrier of R^1
y . E is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
(e * f1) . E is complex real ext-real Element of the carrier of R^1
(f * f1) . E is complex real ext-real Element of the carrier of R^1
|[((e * f1) . E),((f * f1) . E),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*((e * f1) . E)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((e * f1) . E)] is set
{1,((e * f1) . E)} is non empty V195() V196() V197() set
{{1,((e * f1) . E)},{1}} is non empty set
{[1,((e * f1) . E)]} is Relation-like Function-like non empty set
<*((f * f1) . E)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,((f * f1) . E)] is set
{1,((f * f1) . E)} is non empty V195() V196() V197() set
{{1,((f * f1) . E)},{1}} is non empty set
{[1,((f * f1) . E)]} is Relation-like Function-like non empty set
<*((e * f1) . E)*> ^ <*((f * f1) . E)*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*((e * f1) . E)*> ^ <*((f * f1) . E)*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1,2,3,((e * f1) . E),((f * f1) . E),0) is Relation-like Function-like set
(1,2) --> (((e * f1) . E),((f * f1) . E)) is Relation-like REAL -defined Function-like set
1 .--> ((e * f1) . E) is Relation-like REAL -defined {1} -defined the carrier of R^1 -valued Function-like one-to-one constant trivial complex-yielding ext-real-valued real-valued increasing decreasing non-decreasing non-increasing set
{1} --> ((e * f1) . E) is Relation-like {1} -defined the carrier of R^1 -valued {((e * f1) . E)} -valued Function-like constant non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:{1},{((e * f1) . E)}:]
{((e * f1) . E)} is non empty V195() V196() V197() set
[:{1},{((e * f1) . E)}:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:{1},{((e * f1) . E)}:] is set
2 .--> ((f * f1) . E) is Relation-like REAL -defined {2} -defined the carrier of R^1 -valued Function-like one-to-one constant trivial complex-yielding ext-real-valued real-valued increasing decreasing non-decreasing non-increasing set
{2} --> ((f * f1) . E) is Relation-like {2} -defined the carrier of R^1 -valued {((f * f1) . E)} -valued Function-like constant non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:{2},{((f * f1) . E)}:]
{((f * f1) . E)} is non empty V195() V196() V197() set
[:{2},{((f * f1) . E)}:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:{2},{((f * f1) . E)}:] is set
(1 .--> ((e * f1) . E)) +* (2 .--> ((f * f1) . E)) is Relation-like REAL -defined the carrier of R^1 -valued Function-like complex-yielding ext-real-valued real-valued set
3 .--> 0 is Relation-like REAL -defined {3} -defined RAT -valued INT -valued Function-like one-to-one constant trivial complex-yielding ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing Function-yielding V178() set
{3} --> 0 is Relation-like non non-empty {3} -defined RAT -valued INT -valued {0} -valued Function-like constant non empty total quasi_total complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() complex-functions-valued Element of bool [:{3},{0}:]
[:{3},{0}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
bool [:{3},{0}:] is set
((1,2) --> (((e * f1) . E),((f * f1) . E))) +* (3 .--> 0) is Relation-like REAL -defined Function-like set
[: the carrier of I[01], the carrier of (Tunit_circle 3):] is Relation-like set
bool [: the carrier of I[01], the carrier of (Tunit_circle 3):] is set
r is complex real ext-real set
(2 * PI) * r is complex real ext-real Element of REAL
p is complex real ext-real Element of the carrier of I[01]
((2 * PI) * r) * p is complex real ext-real Element of REAL
cos (((2 * PI) * r) * p) is complex real ext-real Element of REAL
sin (((2 * PI) * r) * p) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos (((2 * PI) * r) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos (((2 * PI) * r) * p))] is set
{1,(cos (((2 * PI) * r) * p))} is non empty V195() V196() V197() set
{{1,(cos (((2 * PI) * r) * p))},{1}} is non empty set
{[1,(cos (((2 * PI) * r) * p))]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * r) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * r) * p))] is set
{1,(sin (((2 * PI) * r) * p))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * r) * p))},{1}} is non empty set
{[1,(sin (((2 * PI) * r) * p))]} is Relation-like Function-like non empty set
<*(cos (((2 * PI) * r) * p))*> ^ <*(sin (((2 * PI) * r) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos (((2 * PI) * r) * p))*> ^ <*(sin (((2 * PI) * r) * p))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]| is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]| is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- |[0,0,0]| is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * |[0,0,0]| is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|[0,0,0]| (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[0,0,0]| (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| + (- |[0,0,0]|) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: (|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|,(- |[0,0,0]|)) is Relation-like Function-like set
diffreal .: (|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|,|[0,0,0]|) is Relation-like Function-like set
K254(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|,|[0,0,0]|) is Relation-like Function-like set
K254(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|,|[0,0,0]|) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|.(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|).| is complex real ext-real non negative Element of REAL
sqr (|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|),(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|),(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|)) is Relation-like Function-like set
K254((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|),(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|)) is Relation-like Function-like set
K254((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|),(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| - |[0,0,0]|))) is complex real ext-real Element of REAL
|.|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|.| is complex real ext-real non negative Element of REAL
sqr |[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|,|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|,|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|) is Relation-like Function-like set
K254(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|,|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|) is Relation-like Function-like set
K254(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|,|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr |[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|) is complex real ext-real Element of REAL
sqrt (Sum (sqr |[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]|)) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `1 is complex real ext-real Element of REAL
(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `1) ^2 is complex real ext-real non negative Element of REAL
(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `1) * (|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `1) is complex real ext-real non negative set
|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `2 is complex real ext-real Element of REAL
(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `2) ^2 is complex real ext-real non negative Element of REAL
(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `2) * (|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `2) is complex real ext-real non negative set
((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `1) ^2) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `2) ^2) is complex real ext-real non negative Element of REAL
|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `3 is complex real ext-real Element of REAL
(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `3) ^2 is complex real ext-real non negative Element of REAL
(|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `3) * (|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `3) is complex real ext-real non negative set
(((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `1) ^2) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `2) ^2)) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `3) ^2) is complex real ext-real non negative Element of REAL
sqrt ((((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `1) ^2) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `2) ^2)) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `3) ^2)) is complex real ext-real non negative Element of REAL
(cos (((2 * PI) * r) * p)) ^2 is complex real ext-real non negative Element of REAL
(cos (((2 * PI) * r) * p)) * (cos (((2 * PI) * r) * p)) is complex real ext-real non negative set
((cos (((2 * PI) * r) * p)) ^2) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `2) ^2) is complex real ext-real non negative Element of REAL
(((cos (((2 * PI) * r) * p)) ^2) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `2) ^2)) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `3) ^2) is complex real ext-real non negative Element of REAL
sqrt ((((cos (((2 * PI) * r) * p)) ^2) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `2) ^2)) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `3) ^2)) is complex real ext-real non negative Element of REAL
(sin (((2 * PI) * r) * p)) ^2 is complex real ext-real non negative Element of REAL
(sin (((2 * PI) * r) * p)) * (sin (((2 * PI) * r) * p)) is complex real ext-real non negative set
((cos (((2 * PI) * r) * p)) ^2) + ((sin (((2 * PI) * r) * p)) ^2) is complex real ext-real non negative Element of REAL
(((cos (((2 * PI) * r) * p)) ^2) + ((sin (((2 * PI) * r) * p)) ^2)) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `3) ^2) is complex real ext-real non negative Element of REAL
sqrt ((((cos (((2 * PI) * r) * p)) ^2) + ((sin (((2 * PI) * r) * p)) ^2)) + ((|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| `3) ^2)) is complex real ext-real non negative Element of REAL
Q ^2 is complex real ext-real non negative set
Q * Q is complex ordinal natural real integer ext-real non negative set
(((cos (((2 * PI) * r) * p)) ^2) + ((sin (((2 * PI) * r) * p)) ^2)) + (Q ^2) is complex real ext-real non negative Element of REAL
sqrt ((((cos (((2 * PI) * r) * p)) ^2) + ((sin (((2 * PI) * r) * p)) ^2)) + (Q ^2)) is complex real ext-real non negative Element of REAL
e is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
p is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 3):]
p is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 3):]
f is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 3):]
e is complex real ext-real Element of the carrier of I[01]
p . e is Relation-like Function-like complex-yielding ext-real-valued real-valued set
f . e is Relation-like Function-like complex-yielding ext-real-valued real-valued set
p . e is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
((2 * PI) * r) * e is complex real ext-real Element of REAL
cos (((2 * PI) * r) * e) is complex real ext-real Element of REAL
sin (((2 * PI) * r) * e) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * r) * e)),(sin (((2 * PI) * r) * e)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos (((2 * PI) * r) * e))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos (((2 * PI) * r) * e))] is set
{1,(cos (((2 * PI) * r) * e))} is non empty V195() V196() V197() set
{{1,(cos (((2 * PI) * r) * e))},{1}} is non empty set
{[1,(cos (((2 * PI) * r) * e))]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * r) * e))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * r) * e))] is set
{1,(sin (((2 * PI) * r) * e))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * r) * e))},{1}} is non empty set
{[1,(sin (((2 * PI) * r) * e))]} is Relation-like Function-like non empty set
<*(cos (((2 * PI) * r) * e))*> ^ <*(sin (((2 * PI) * r) * e))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos (((2 * PI) * r) * e))*> ^ <*(sin (((2 * PI) * r) * e))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
f . e is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
r is complex real ext-real set
(r) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 3):]
R^1 r is complex real ext-real Element of the carrier of K708()
ExtendInt r is Relation-like the carrier of I[01] -defined the carrier of K708() -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued Path of R^1 0, R^1 r
() * (ExtendInt r) is Relation-like the carrier of I[01] -defined the carrier of I[01] -defined the carrier of (Tunit_circle 3) -valued the carrier of (Tunit_circle 3) -valued Function-like non empty total total quasi_total quasi_total quasi_total continuous Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of () . (R^1 0),() . (R^1 r)
() . (R^1 0) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
() . (R^1 r) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
p is complex real ext-real Element of the carrier of I[01]
(r) . p is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(() * (ExtendInt r)) . p is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(ExtendInt r) . p is complex real ext-real Element of the carrier of K708()
r * p is complex real ext-real set
(r) . p is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
(2 * PI) * r is complex real ext-real Element of REAL
((2 * PI) * r) * p is complex real ext-real Element of REAL
cos (((2 * PI) * r) * p) is complex real ext-real Element of REAL
sin (((2 * PI) * r) * p) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * r) * p)),(sin (((2 * PI) * r) * p)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos (((2 * PI) * r) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos (((2 * PI) * r) * p))] is set
{1,(cos (((2 * PI) * r) * p))} is non empty V195() V196() V197() set
{{1,(cos (((2 * PI) * r) * p))},{1}} is non empty set
{[1,(cos (((2 * PI) * r) * p))]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * r) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * r) * p))] is set
{1,(sin (((2 * PI) * r) * p))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * r) * p))},{1}} is non empty set
{[1,(sin (((2 * PI) * r) * p))]} is Relation-like Function-like non empty set
<*(cos (((2 * PI) * r) * p))*> ^ <*(sin (((2 * PI) * r) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos (((2 * PI) * r) * p))*> ^ <*(sin (((2 * PI) * r) * p))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(2 * PI) * ((ExtendInt r) . p) is complex real ext-real Element of REAL
cos ((2 * PI) * ((ExtendInt r) . p)) is complex real ext-real Element of REAL
sin ((2 * PI) * ((ExtendInt r) . p)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * ((ExtendInt r) . p))),(sin ((2 * PI) * ((ExtendInt r) . p))),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos ((2 * PI) * ((ExtendInt r) . p)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos ((2 * PI) * ((ExtendInt r) . p)))] is set
{1,(cos ((2 * PI) * ((ExtendInt r) . p)))} is non empty V195() V196() V197() set
{{1,(cos ((2 * PI) * ((ExtendInt r) . p)))},{1}} is non empty set
{[1,(cos ((2 * PI) * ((ExtendInt r) . p)))]} is Relation-like Function-like non empty set
<*(sin ((2 * PI) * ((ExtendInt r) . p)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin ((2 * PI) * ((ExtendInt r) . p)))] is set
{1,(sin ((2 * PI) * ((ExtendInt r) . p)))} is non empty V195() V196() V197() set
{{1,(sin ((2 * PI) * ((ExtendInt r) . p)))},{1}} is non empty set
{[1,(sin ((2 * PI) * ((ExtendInt r) . p)))]} is Relation-like Function-like non empty set
<*(cos ((2 * PI) * ((ExtendInt r) . p)))*> ^ <*(sin ((2 * PI) * ((ExtendInt r) . p)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos ((2 * PI) * ((ExtendInt r) . p)))*> ^ <*(sin ((2 * PI) * ((ExtendInt r) . p)))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
() . ((ExtendInt r) . p) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
(() * (ExtendInt r)) . p is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
r is complex real integer ext-real set
(r) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 3):]
(r) . 0 is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(r) . 1 is Relation-like Function-like complex-yielding ext-real-valued real-valued set
R^1 r is complex real ext-real Element of the carrier of K708()
ExtendInt r is Relation-like the carrier of I[01] -defined the carrier of K708() -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued Path of R^1 0, R^1 r
() * (ExtendInt r) is Relation-like the carrier of I[01] -defined the carrier of I[01] -defined the carrier of (Tunit_circle 3) -valued the carrier of (Tunit_circle 3) -valued Function-like non empty total total quasi_total quasi_total quasi_total continuous Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of () . (R^1 0),() . (R^1 r)
() . (R^1 0) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
() . (R^1 r) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
(2 * PI) * r is complex real ext-real Element of REAL
j0 is complex real ext-real Element of the carrier of I[01]
((2 * PI) * r) * j0 is complex real ext-real Element of REAL
cos (((2 * PI) * r) * j0) is complex real ext-real Element of REAL
sin (((2 * PI) * r) * j0) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * r) * j0)),(sin (((2 * PI) * r) * j0)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos (((2 * PI) * r) * j0))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos (((2 * PI) * r) * j0))] is set
{1,(cos (((2 * PI) * r) * j0))} is non empty V195() V196() V197() set
{{1,(cos (((2 * PI) * r) * j0))},{1}} is non empty set
{[1,(cos (((2 * PI) * r) * j0))]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * r) * j0))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * r) * j0))] is set
{1,(sin (((2 * PI) * r) * j0))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * r) * j0))},{1}} is non empty set
{[1,(sin (((2 * PI) * r) * j0))]} is Relation-like Function-like non empty set
<*(cos (((2 * PI) * r) * j0))*> ^ <*(sin (((2 * PI) * r) * j0))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos (((2 * PI) * r) * j0))*> ^ <*(sin (((2 * PI) * r) * j0))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
j1 is complex real ext-real Element of the carrier of I[01]
((2 * PI) * r) * j1 is complex real ext-real Element of REAL
cos (((2 * PI) * r) * j1) is complex real ext-real Element of REAL
sin (((2 * PI) * r) * j1) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * r) * j1)),(sin (((2 * PI) * r) * j1)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos (((2 * PI) * r) * j1))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos (((2 * PI) * r) * j1))] is set
{1,(cos (((2 * PI) * r) * j1))} is non empty V195() V196() V197() set
{{1,(cos (((2 * PI) * r) * j1))},{1}} is non empty set
{[1,(cos (((2 * PI) * r) * j1))]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * r) * j1))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * r) * j1))] is set
{1,(sin (((2 * PI) * r) * j1))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * r) * j1))},{1}} is non empty set
{[1,(sin (((2 * PI) * r) * j1))]} is Relation-like Function-like non empty set
<*(cos (((2 * PI) * r) * j1))*> ^ <*(sin (((2 * PI) * r) * j1))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos (((2 * PI) * r) * j1))*> ^ <*(sin (((2 * PI) * r) * j1))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((2 * PI) * r) + Q is complex real ext-real Element of REAL
sin (((2 * PI) * r) + Q) is complex real ext-real Element of REAL
|[(cos 0),(sin (((2 * PI) * r) + Q)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos 0)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos 0)] is set
{1,(cos 0)} is non empty V195() V196() V197() set
{{1,(cos 0)},{1}} is non empty set
{[1,(cos 0)]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * r) + Q))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * r) + Q))] is set
{1,(sin (((2 * PI) * r) + Q))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * r) + Q))},{1}} is non empty set
{[1,(sin (((2 * PI) * r) + Q))]} is Relation-like Function-like non empty set
<*(cos 0)*> ^ <*(sin (((2 * PI) * r) + Q))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos 0)*> ^ <*(sin (((2 * PI) * r) + Q))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
r is complex real integer ext-real set
(r) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of (),()
p is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of (),()
r is complex ordinal natural real integer ext-real non negative set
TOP-REAL r is non empty TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
0. (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
the ZeroF of (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
p is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
|.p.| is complex real ext-real non negative Element of REAL
sqr p is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (p,p) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (p,p) is Relation-like Function-like set
K254(p,p) is Relation-like Function-like set
K254(p,p) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
p (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr p) is complex real ext-real Element of REAL
sqrt (Sum (sqr p)) is complex real ext-real Element of REAL
p (/) |.p.| is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
1 / |.p.| is complex real ext-real non negative set
|.p.| " is complex real ext-real non negative set
1 * (|.p.| ") is complex real ext-real non negative set
(1 / |.p.|) * p is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 / |.p.|) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((1 / |.p.|),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (1 / |.p.|) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 / |.p.|)} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 / |.p.|)}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 / |.p.|)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 / |.p.|)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
p (#) ((1 / |.p.|) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|.(p (/) |.p.|).| is complex real ext-real non negative Element of REAL
sqr (p (/) |.p.|) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((p (/) |.p.|),(p (/) |.p.|)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((p (/) |.p.|),(p (/) |.p.|)) is Relation-like Function-like set
K254((p (/) |.p.|),(p (/) |.p.|)) is Relation-like Function-like set
K254((p (/) |.p.|),(p (/) |.p.|)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(p (/) |.p.|) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (p (/) |.p.|)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (p (/) |.p.|))) is complex real ext-real Element of REAL
|.p.| ^2 is complex real ext-real non negative Element of REAL
|.p.| * |.p.| is complex real ext-real non negative set
sqr p is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
Sum (sqr p) is complex real ext-real set
(sqr p) (/) (Sum (sqr p)) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
1 / (Sum (sqr p)) is complex real ext-real set
(Sum (sqr p)) " is complex real ext-real set
1 * ((Sum (sqr p)) ") is complex real ext-real set
(1 / (Sum (sqr p))) * (sqr p) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 / (Sum (sqr p))) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((1 / (Sum (sqr p))),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (1 / (Sum (sqr p))) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 / (Sum (sqr p)))} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 / (Sum (sqr p)))}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 / (Sum (sqr p)))),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 / (Sum (sqr p)))),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(sqr p) (#) ((1 / (Sum (sqr p))) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum ((sqr p) (/) (Sum (sqr p))) is complex real ext-real set
r is complex ordinal natural real integer ext-real non negative set
TOP-REAL r is non empty TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
p is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
0. (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
the ZeroF of (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
|.p.| is complex real ext-real non negative Element of REAL
sqr p is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (p,p) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() .: (p,p) is Relation-like Function-like set
K254(p,p) is Relation-like Function-like set
K254(p,p) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
p (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr p) is complex real ext-real Element of REAL
sqrt (Sum (sqr p)) is complex real ext-real Element of REAL
p (/) |.p.| is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
1 / |.p.| is complex real ext-real non negative set
|.p.| " is complex real ext-real non negative set
1 * (|.p.| ") is complex real ext-real non negative set
(1 / |.p.|) * p is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 / |.p.|) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((1 / |.p.|),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (1 / |.p.|) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 / |.p.|)} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 / |.p.|)}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 / |.p.|)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 / |.p.|)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
p (#) ((1 / |.p.|) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Tcircle ((0. (TOP-REAL r)),1) is TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Sphere ((0. (TOP-REAL r)),1) is functional closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is V74() set
(TOP-REAL r) | (Sphere ((0. (TOP-REAL r)),1)) is strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
the carrier of (Tcircle ((0. (TOP-REAL r)),1)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
(p (/) |.p.|) - (0. (TOP-REAL r)) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(p (/) |.p.|) - (0. (TOP-REAL r)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (0. (TOP-REAL r)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (0. (TOP-REAL r)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(0. (TOP-REAL r)) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(0. (TOP-REAL r)) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p (/) |.p.|) + (- (0. (TOP-REAL r))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((p (/) |.p.|),(- (0. (TOP-REAL r)))) is Relation-like Function-like set
diffreal .: ((p (/) |.p.|),(0. (TOP-REAL r))) is Relation-like Function-like set
K254((p (/) |.p.|),(0. (TOP-REAL r))) is Relation-like Function-like set
K254((p (/) |.p.|),(0. (TOP-REAL r))) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|.((p (/) |.p.|) - (0. (TOP-REAL r))).| is complex real ext-real non negative Element of REAL
sqr ((p (/) |.p.|) - (0. (TOP-REAL r))) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (((p (/) |.p.|) - (0. (TOP-REAL r))),((p (/) |.p.|) - (0. (TOP-REAL r)))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (((p (/) |.p.|) - (0. (TOP-REAL r))),((p (/) |.p.|) - (0. (TOP-REAL r)))) is Relation-like Function-like set
K254(((p (/) |.p.|) - (0. (TOP-REAL r))),((p (/) |.p.|) - (0. (TOP-REAL r)))) is Relation-like Function-like set
K254(((p (/) |.p.|) - (0. (TOP-REAL r))),((p (/) |.p.|) - (0. (TOP-REAL r)))) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((p (/) |.p.|) - (0. (TOP-REAL r))) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr ((p (/) |.p.|) - (0. (TOP-REAL r)))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((p (/) |.p.|) - (0. (TOP-REAL r))))) is complex real ext-real Element of REAL
|.(p (/) |.p.|).| is complex real ext-real non negative Element of REAL
sqr (p (/) |.p.|) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((p (/) |.p.|),(p (/) |.p.|)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((p (/) |.p.|),(p (/) |.p.|)) is Relation-like Function-like set
K254((p (/) |.p.|),(p (/) |.p.|)) is Relation-like Function-like set
K254((p (/) |.p.|),(p (/) |.p.|)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(p (/) |.p.|) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (p (/) |.p.|)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (p (/) |.p.|))) is complex real ext-real Element of REAL
r is non empty complex ordinal natural real integer ext-real positive non negative set
r + 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
TOP-REAL (r + 1) is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
0. (TOP-REAL (r + 1)) is Relation-like NAT -defined Function-like V32() V39(r + 1) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
the carrier of (TOP-REAL (r + 1)) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (r + 1)) is Relation-like NAT -defined Function-like V32() V39(r + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
Tcircle ((0. (TOP-REAL (r + 1))),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
Sphere ((0. (TOP-REAL (r + 1))),1) is functional non empty closed V214( TOP-REAL (r + 1)) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (r + 1))
bool the carrier of (TOP-REAL (r + 1)) is V74() set
(TOP-REAL (r + 1)) | (Sphere ((0. (TOP-REAL (r + 1))),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
TOP-REAL r is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):] is set
Tunit_circle (r + 1) is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
Tunit_circle r is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
0. (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
the ZeroF of (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
Tcircle ((0. (TOP-REAL r)),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Sphere ((0. (TOP-REAL r)),1) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is V74() set
(TOP-REAL r) | (Sphere ((0. (TOP-REAL r)),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
the carrier of (Tunit_circle (r + 1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
the carrier of (Tunit_circle r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):] is Relation-like set
bool [: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):] is set
p is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):]
h is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
- h is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * h is Relation-like Function-like complex-yielding ext-real-valued real-valued set
p . h is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
x is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
p . x is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(p . h) - (p . x) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(p . h) - (p . x) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (p . x) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (p . x) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(p . x) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p . x) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p . h) + (- (p . x)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((p . h),(- (p . x))) is Relation-like Function-like set
diffreal .: ((p . h),(p . x)) is Relation-like Function-like set
K254((p . h),(p . x)) is Relation-like Function-like set
K254((p . h),(p . x)) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(r,((p . h) - (p . x))) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL r)),1))
the carrier of (Tcircle ((0. (TOP-REAL r)),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
y is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle r)
hx is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
p . hx is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(p . h) - (p . hx) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(p . h) - (p . hx) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (p . hx) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (p . hx) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p . hx) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p . hx) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p . h) + (- (p . hx)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((p . h),(- (p . hx))) is Relation-like Function-like set
diffreal .: ((p . h),(p . hx)) is Relation-like Function-like set
K254((p . h),(p . hx)) is Relation-like Function-like set
K254((p . h),(p . hx)) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(r,((p . h) - (p . hx))) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL r)),1))
[: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (Tunit_circle r):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (Tunit_circle r):] is set
h is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of (Tunit_circle r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (Tunit_circle r):]
h is Relation-like the carrier of (Tunit_circle (r + 1)) -defined the carrier of (Tunit_circle r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):]
x is Relation-like the carrier of (Tunit_circle (r + 1)) -defined the carrier of (Tunit_circle r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):]
y is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle (r + 1))
h . y is Relation-like Function-like complex-yielding ext-real-valued real-valued set
x . y is Relation-like Function-like complex-yielding ext-real-valued real-valued set
- y is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * y is Relation-like Function-like complex-yielding ext-real-valued real-valued set
h . y is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle r)
hx is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
p . hx is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
hy is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
p . hy is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(p . hx) - (p . hy) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(p . hx) - (p . hy) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (p . hy) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (p . hy) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(p . hy) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p . hy) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p . hx) + (- (p . hy)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((p . hx),(- (p . hy))) is Relation-like Function-like set
diffreal .: ((p . hx),(p . hy)) is Relation-like Function-like set
K254((p . hx),(p . hy)) is Relation-like Function-like set
K254((p . hx),(p . hy)) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(r,((p . hx) - (p . hy))) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL r)),1))
the carrier of (Tcircle ((0. (TOP-REAL r)),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
x . y is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle r)
r is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
p is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
f is set
{r} is functional non empty complex-functions-membered set
CircleMap " {r} is V195() V196() V197() Element of bool the carrier of K708()
[: the carrier of I[01], the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of I[01], the carrier of R^1:] is set
e is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of r,p
f1 is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
f1 . 0 is complex real ext-real set
CircleMap * f1 is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
[: the carrier of I[01], the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of I[01], the carrier of (Tunit_circle 2):] is set
h is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
h . 0 is complex real ext-real set
CircleMap * h is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
r is complex ordinal natural real integer ext-real non negative set
TOP-REAL r is non empty TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is complex ordinal natural real integer ext-real non negative set
TOP-REAL r is non empty TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is complex ordinal natural real integer ext-real non negative set
TOP-REAL r is non empty TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
f is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
e is complex real ext-real set
Tcircle (f,e) is TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Sphere (f,e) is functional closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is V74() set
(TOP-REAL r) | (Sphere (f,e)) is strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
the carrier of (Tcircle (f,e)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
p is complex ordinal natural real integer ext-real non negative set
TOP-REAL p is non empty TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL p) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle (f,e)), the carrier of (TOP-REAL p):] is Relation-like set
bool [: the carrier of (Tcircle (f,e)), the carrier of (TOP-REAL p):] is set
r is complex ordinal natural real integer ext-real non negative set
TOP-REAL r is non empty TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
f is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
e is complex real ext-real set
Tcircle (f,e) is TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Sphere (f,e) is functional closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is V74() set
(TOP-REAL r) | (Sphere (f,e)) is strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
the carrier of (Tcircle (f,e)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
p is complex ordinal natural real integer ext-real non negative set
TOP-REAL p is non empty TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL p) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle (f,e)), the carrier of (TOP-REAL p):] is Relation-like set
bool [: the carrier of (Tcircle (f,e)), the carrier of (TOP-REAL p):] is set
r is non empty complex ordinal natural real integer ext-real positive non negative set
TOP-REAL r is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
0. (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
the ZeroF of (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
p is complex real ext-real non negative set
Tcircle ((0. (TOP-REAL r)),p) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Sphere ((0. (TOP-REAL r)),p) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is V74() set
(TOP-REAL r) | (Sphere ((0. (TOP-REAL r)),p)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
the carrier of (Tcircle ((0. (TOP-REAL r)),p)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
f is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
- f is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
- f is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(- 1) * f is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
f (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
e is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL r)),p))
- e is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * e is Relation-like Function-like complex-yielding ext-real-valued real-valued set
LSeg (f,(- f)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
{ (((1 - b₁) * f) + (b₁ * (- f))) where b₁ is complex real ext-real Element of REAL : ( 0 <= b₁ & b₁ <= 1 ) } is set
1 - (1 / 2) is complex real ext-real Element of REAL
- (1 / 2) is non empty complex real ext-real non positive negative set
1 + (- (1 / 2)) is complex real ext-real set
(1 - (1 / 2)) * f is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(1 - (1 / 2)) * f is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(1 - (1 / 2)) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((1 - (1 / 2)),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (1 - (1 / 2)) is Relation-like dom (id REAL) -defined REAL -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 - (1 / 2))} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 - (1 / 2))}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 - (1 / 2))),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 - (1 / 2))),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
f (#) ((1 - (1 / 2)) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 / 2) * (- f) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(1 / 2) * (- f) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(1 / 2) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((1 / 2),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (1 / 2) is Relation-like non-empty dom (id REAL) -defined COMPLEX -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 / 2)} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 / 2)}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 / 2)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 / 2)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(- f) (#) ((1 / 2) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((1 - (1 / 2)) * f) + ((1 / 2) * (- f)) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((1 - (1 / 2)) * f) + ((1 / 2) * (- f)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((1 - (1 / 2)) * f),((1 / 2) * (- f))) is Relation-like Function-like set
K254(((1 - (1 / 2)) * f),((1 / 2) * (- f))) is Relation-like Function-like set
K254(((1 - (1 / 2)) * f),((1 / 2) * (- f))) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(1 / 2) * f is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(1 / 2) * f is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
f (#) ((1 / 2) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- ((1 / 2) * f) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
- ((1 / 2) * f) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(- 1) * ((1 / 2) * f) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((1 / 2) * f) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((1 / 2) * f) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((1 / 2) * f) + (- ((1 / 2) * f)) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((1 / 2) * f) + (- ((1 / 2) * f)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((1 / 2) * f),(- ((1 / 2) * f))) is Relation-like Function-like set
K254(((1 / 2) * f),(- ((1 / 2) * f))) is Relation-like Function-like set
K254(((1 / 2) * f),(- ((1 / 2) * f))) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
r is non empty complex ordinal natural real integer ext-real positive non negative set
TOP-REAL r is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
0. (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
the ZeroF of (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
f is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
e is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
f1 is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
p is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
h is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
x is non empty complex real ext-real positive non negative set
(r,p,x) is Relation-like the carrier of (Tunit_circle r) -defined the carrier of (Tcircle (p,x)) -valued Function-like one-to-one non empty total quasi_total onto bijective continuous Function-yielding V178() being_homeomorphism complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle r), the carrier of (Tcircle (p,x)):]
Tunit_circle r is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Tcircle ((0. (TOP-REAL r)),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Sphere ((0. (TOP-REAL r)),1) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is V74() set
(TOP-REAL r) | (Sphere ((0. (TOP-REAL r)),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
the carrier of (Tunit_circle r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
Tcircle (p,x) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Sphere (p,x) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
(TOP-REAL r) | (Sphere (p,x)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
the carrier of (Tcircle (p,x)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tunit_circle r), the carrier of (Tcircle (p,x)):] is Relation-like set
bool [: the carrier of (Tunit_circle r), the carrier of (Tcircle (p,x)):] is set
(r,p,x) . f is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(r,p,x) . e is Relation-like Function-like complex-yielding ext-real-valued real-valued set
the carrier of (Tcircle ((0. (TOP-REAL r)),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
LSeg (f1,h) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
{ (((1 - b₁) * f1) + (b₁ * h)) where b₁ is complex real ext-real Element of REAL : ( 0 <= b₁ & b₁ <= 1 ) } is set
LSeg (f,e) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
{ (((1 - b₁) * f) + (b₁ * e)) where b₁ is complex real ext-real Element of REAL : ( 0 <= b₁ & b₁ <= 1 ) } is set
hx is complex real ext-real Element of REAL
1 - hx is complex real ext-real Element of REAL
- hx is complex real ext-real set
1 + (- hx) is complex real ext-real set
(1 - hx) * f is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(1 - hx) * f is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(1 - hx) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((1 - hx),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (1 - hx) is Relation-like dom (id REAL) -defined REAL -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 - hx)} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 - hx)}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 - hx)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 - hx)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
f (#) ((1 - hx) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
hx * e is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
hx * e is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
hx multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] (hx,(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> hx is Relation-like dom (id REAL) -defined REAL -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{hx} is non empty V195() V196() V197() set
[:(dom (id REAL)),{hx}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> hx),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> hx),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
e (#) (hx multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((1 - hx) * f) + (hx * e) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((1 - hx) * f) + (hx * e) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((1 - hx) * f),(hx * e)) is Relation-like Function-like set
K254(((1 - hx) * f),(hx * e)) is Relation-like Function-like set
K254(((1 - hx) * f),(hx * e)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(1 - hx) * f1 is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(1 - hx) * f1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
f1 (#) ((1 - hx) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
x * f is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
x * f is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
x multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] (x,(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> x is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{x} is non empty V195() V196() V197() set
[:(dom (id REAL)),{x}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> x),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> x),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
f (#) (x multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(x * f) + p is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(x * f) + p is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((x * f),p) is Relation-like Function-like set
K254((x * f),p) is Relation-like Function-like set
K254((x * f),p) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(1 - hx) * ((x * f) + p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(1 - hx) * ((x * f) + p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
((x * f) + p) (#) ((1 - hx) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 - hx) * (x * f) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(1 - hx) * (x * f) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(x * f) (#) ((1 - hx) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 - hx) * p is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(1 - hx) * p is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
p (#) ((1 - hx) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((1 - hx) * (x * f)) + ((1 - hx) * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((1 - hx) * (x * f)) + ((1 - hx) * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((1 - hx) * (x * f)),((1 - hx) * p)) is Relation-like Function-like set
K254(((1 - hx) * (x * f)),((1 - hx) * p)) is Relation-like Function-like set
K254(((1 - hx) * (x * f)),((1 - hx) * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
x * (1 - hx) is complex real ext-real Element of REAL
(x * (1 - hx)) * f is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(x * (1 - hx)) * f is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(x * (1 - hx)) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((x * (1 - hx)),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (x * (1 - hx)) is Relation-like dom (id REAL) -defined REAL -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(x * (1 - hx))} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(x * (1 - hx))}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (x * (1 - hx))),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (x * (1 - hx))),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
f (#) ((x * (1 - hx)) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((x * (1 - hx)) * f) + ((1 - hx) * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((x * (1 - hx)) * f) + ((1 - hx) * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((x * (1 - hx)) * f),((1 - hx) * p)) is Relation-like Function-like set
K254(((x * (1 - hx)) * f),((1 - hx) * p)) is Relation-like Function-like set
K254(((x * (1 - hx)) * f),((1 - hx) * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
x * ((1 - hx) * f) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
x * ((1 - hx) * f) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
((1 - hx) * f) (#) (x multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(x * ((1 - hx) * f)) + ((1 - hx) * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(x * ((1 - hx) * f)) + ((1 - hx) * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((x * ((1 - hx) * f)),((1 - hx) * p)) is Relation-like Function-like set
K254((x * ((1 - hx) * f)),((1 - hx) * p)) is Relation-like Function-like set
K254((x * ((1 - hx) * f)),((1 - hx) * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
hx * h is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
hx * h is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
h (#) (hx multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
x * e is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
x * e is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
e (#) (x multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(x * e) + p is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(x * e) + p is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((x * e),p) is Relation-like Function-like set
K254((x * e),p) is Relation-like Function-like set
K254((x * e),p) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
hx * ((x * e) + p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
hx * ((x * e) + p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
((x * e) + p) (#) (hx multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
hx * (x * e) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
hx * (x * e) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(x * e) (#) (hx multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
hx * p is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
hx * p is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
p (#) (hx multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(hx * (x * e)) + (hx * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(hx * (x * e)) + (hx * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((hx * (x * e)),(hx * p)) is Relation-like Function-like set
K254((hx * (x * e)),(hx * p)) is Relation-like Function-like set
K254((hx * (x * e)),(hx * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
x * hx is complex real ext-real Element of REAL
(x * hx) * e is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(x * hx) * e is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(x * hx) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((x * hx),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (x * hx) is Relation-like dom (id REAL) -defined REAL -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(x * hx)} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(x * hx)}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (x * hx)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (x * hx)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
e (#) ((x * hx) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((x * hx) * e) + (hx * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((x * hx) * e) + (hx * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((x * hx) * e),(hx * p)) is Relation-like Function-like set
K254(((x * hx) * e),(hx * p)) is Relation-like Function-like set
K254(((x * hx) * e),(hx * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
x * (hx * e) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
x * (hx * e) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(hx * e) (#) (x multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(x * (hx * e)) + (hx * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(x * (hx * e)) + (hx * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((x * (hx * e)),(hx * p)) is Relation-like Function-like set
K254((x * (hx * e)),(hx * p)) is Relation-like Function-like set
K254((x * (hx * e)),(hx * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((1 - hx) * f1) + (hx * h) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((1 - hx) * f1) + (hx * h) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((1 - hx) * f1),(hx * h)) is Relation-like Function-like set
K254(((1 - hx) * f1),(hx * h)) is Relation-like Function-like set
K254(((1 - hx) * f1),(hx * h)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((x * ((1 - hx) * f)) + ((1 - hx) * p)) + (x * (hx * e)) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((x * ((1 - hx) * f)) + ((1 - hx) * p)) + (x * (hx * e)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((x * ((1 - hx) * f)) + ((1 - hx) * p)),(x * (hx * e))) is Relation-like Function-like set
K254(((x * ((1 - hx) * f)) + ((1 - hx) * p)),(x * (hx * e))) is Relation-like Function-like set
K254(((x * ((1 - hx) * f)) + ((1 - hx) * p)),(x * (hx * e))) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(((x * ((1 - hx) * f)) + ((1 - hx) * p)) + (x * (hx * e))) + (hx * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(((x * ((1 - hx) * f)) + ((1 - hx) * p)) + (x * (hx * e))) + (hx * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((((x * ((1 - hx) * f)) + ((1 - hx) * p)) + (x * (hx * e))),(hx * p)) is Relation-like Function-like set
K254((((x * ((1 - hx) * f)) + ((1 - hx) * p)) + (x * (hx * e))),(hx * p)) is Relation-like Function-like set
K254((((x * ((1 - hx) * f)) + ((1 - hx) * p)) + (x * (hx * e))),(hx * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(x * ((1 - hx) * f)) + (x * (hx * e)) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(x * ((1 - hx) * f)) + (x * (hx * e)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((x * ((1 - hx) * f)),(x * (hx * e))) is Relation-like Function-like set
K254((x * ((1 - hx) * f)),(x * (hx * e))) is Relation-like Function-like set
K254((x * ((1 - hx) * f)),(x * (hx * e))) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((x * ((1 - hx) * f)) + (x * (hx * e))) + ((1 - hx) * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((x * ((1 - hx) * f)) + (x * (hx * e))) + ((1 - hx) * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((x * ((1 - hx) * f)) + (x * (hx * e))),((1 - hx) * p)) is Relation-like Function-like set
K254(((x * ((1 - hx) * f)) + (x * (hx * e))),((1 - hx) * p)) is Relation-like Function-like set
K254(((x * ((1 - hx) * f)) + (x * (hx * e))),((1 - hx) * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(((x * ((1 - hx) * f)) + (x * (hx * e))) + ((1 - hx) * p)) + (hx * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(((x * ((1 - hx) * f)) + (x * (hx * e))) + ((1 - hx) * p)) + (hx * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((((x * ((1 - hx) * f)) + (x * (hx * e))) + ((1 - hx) * p)),(hx * p)) is Relation-like Function-like set
K254((((x * ((1 - hx) * f)) + (x * (hx * e))) + ((1 - hx) * p)),(hx * p)) is Relation-like Function-like set
K254((((x * ((1 - hx) * f)) + (x * (hx * e))) + ((1 - hx) * p)),(hx * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
x * (((1 - hx) * f) + (hx * e)) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
x * (((1 - hx) * f) + (hx * e)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(((1 - hx) * f) + (hx * e)) (#) (x multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(x * (((1 - hx) * f) + (hx * e))) + ((1 - hx) * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(x * (((1 - hx) * f) + (hx * e))) + ((1 - hx) * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((x * (((1 - hx) * f) + (hx * e))),((1 - hx) * p)) is Relation-like Function-like set
K254((x * (((1 - hx) * f) + (hx * e))),((1 - hx) * p)) is Relation-like Function-like set
K254((x * (((1 - hx) * f) + (hx * e))),((1 - hx) * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((x * (((1 - hx) * f) + (hx * e))) + ((1 - hx) * p)) + (hx * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((x * (((1 - hx) * f) + (hx * e))) + ((1 - hx) * p)) + (hx * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((x * (((1 - hx) * f) + (hx * e))) + ((1 - hx) * p)),(hx * p)) is Relation-like Function-like set
K254(((x * (((1 - hx) * f) + (hx * e))) + ((1 - hx) * p)),(hx * p)) is Relation-like Function-like set
K254(((x * (((1 - hx) * f) + (hx * e))) + ((1 - hx) * p)),(hx * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(0. (TOP-REAL r)) + ((1 - hx) * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(0. (TOP-REAL r)) + ((1 - hx) * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: ((0. (TOP-REAL r)),((1 - hx) * p)) is Relation-like Function-like set
K254((0. (TOP-REAL r)),((1 - hx) * p)) is Relation-like Function-like set
K254((0. (TOP-REAL r)),((1 - hx) * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((0. (TOP-REAL r)) + ((1 - hx) * p)) + (hx * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((0. (TOP-REAL r)) + ((1 - hx) * p)) + (hx * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((0. (TOP-REAL r)) + ((1 - hx) * p)),(hx * p)) is Relation-like Function-like set
K254(((0. (TOP-REAL r)) + ((1 - hx) * p)),(hx * p)) is Relation-like Function-like set
K254(((0. (TOP-REAL r)) + ((1 - hx) * p)),(hx * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((1 - hx) * p) + (hx * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((1 - hx) * p) + (hx * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((1 - hx) * p),(hx * p)) is Relation-like Function-like set
K254(((1 - hx) * p),(hx * p)) is Relation-like Function-like set
K254(((1 - hx) * p),(hx * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
1 * p is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
1 * p is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
1 multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] (1,(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> 1 is Relation-like non-empty dom (id REAL) -defined RAT -valued INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued natural-valued set
[:(dom (id REAL)),{1}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
K254(((dom (id REAL)) --> 1),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> 1),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
p (#) (1 multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 * p) - (hx * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(1 * p) - (hx * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (hx * p) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (hx * p) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(hx * p) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(hx * p) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 * p) + (- (hx * p)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((1 * p),(- (hx * p))) is Relation-like Function-like set
diffreal .: ((1 * p),(hx * p)) is Relation-like Function-like set
K254((1 * p),(hx * p)) is Relation-like Function-like set
K254((1 * p),(hx * p)) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((1 * p) - (hx * p)) + (hx * p) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((1 * p) - (hx * p)) + (hx * p) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
K143() .: (((1 * p) - (hx * p)),(hx * p)) is Relation-like Function-like set
K254(((1 * p) - (hx * p)),(hx * p)) is Relation-like Function-like set
K254(((1 * p) - (hx * p)),(hx * p)) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
r is non empty complex ordinal natural real integer ext-real positive non negative set
r + 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
TOP-REAL (r + 1) is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
0. (TOP-REAL (r + 1)) is Relation-like NAT -defined Function-like V32() V39(r + 1) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
the carrier of (TOP-REAL (r + 1)) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (r + 1)) is Relation-like NAT -defined Function-like V32() V39(r + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
Tcircle ((0. (TOP-REAL (r + 1))),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
Sphere ((0. (TOP-REAL (r + 1))),1) is functional non empty closed V214( TOP-REAL (r + 1)) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (r + 1))
bool the carrier of (TOP-REAL (r + 1)) is V74() set
(TOP-REAL (r + 1)) | (Sphere ((0. (TOP-REAL (r + 1))),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
TOP-REAL r is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):] is set
0. (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
the ZeroF of (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
f is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):]
e is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
f . e is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
- e is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * e is Relation-like Function-like complex-yielding ext-real-valued real-valued set
f . (- e) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(f . e) - (f . (- e)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
- (f . (- e)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * (f . (- e)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(f . e) + (- (f . (- e))) is Relation-like NAT -defined Function-like complex-yielding ext-real-valued real-valued set
dom f is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) is V74() set
f1 is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
y is Relation-like NAT -defined Function-like V32() V39(r + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
- y is Relation-like NAT -defined Function-like V32() V39(r + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
- y is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(- 1) * y is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
y (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
y (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
dom f is non empty set
f . y is Relation-like Function-like complex-yielding ext-real-valued real-valued set
f . (- y) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
h is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
f . h is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
f . f1 is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
x is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
f . x is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(f . h) - (f . f1) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(f . h) - (f . f1) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (f . f1) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (f . f1) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . f1) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . f1) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . h) + (- (f . f1)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((f . h),(- (f . f1))) is Relation-like Function-like set
diffreal .: ((f . h),(f . f1)) is Relation-like Function-like set
K254((f . h),(f . f1)) is Relation-like Function-like set
K254((f . h),(f . f1)) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
r is non empty complex ordinal natural real integer ext-real positive non negative set
r + 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
TOP-REAL (r + 1) is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
0. (TOP-REAL (r + 1)) is Relation-like NAT -defined Function-like V32() V39(r + 1) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
the carrier of (TOP-REAL (r + 1)) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (r + 1)) is Relation-like NAT -defined Function-like V32() V39(r + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
Tcircle ((0. (TOP-REAL (r + 1))),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
Sphere ((0. (TOP-REAL (r + 1))),1) is functional non empty closed V214( TOP-REAL (r + 1)) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (r + 1))
bool the carrier of (TOP-REAL (r + 1)) is V74() set
(TOP-REAL (r + 1)) | (Sphere ((0. (TOP-REAL (r + 1))),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
TOP-REAL r is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):] is set
f is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):]
(r,f) is Relation-like the carrier of (Tunit_circle (r + 1)) -defined the carrier of (Tunit_circle r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):]
Tunit_circle (r + 1) is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
the carrier of (Tunit_circle (r + 1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
Tunit_circle r is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
0. (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
the ZeroF of (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
Tcircle ((0. (TOP-REAL r)),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Sphere ((0. (TOP-REAL r)),1) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is V74() set
(TOP-REAL r) | (Sphere ((0. (TOP-REAL r)),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
the carrier of (Tunit_circle r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):] is Relation-like set
bool [: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):] is set
f1 is Relation-like Function-like complex-yielding set
dom (r,f) is non empty set
- f1 is Relation-like Function-like complex-yielding set
(- 1) * f1 is Relation-like Function-like complex-yielding set
h is Relation-like Function-like complex-yielding set
(r,f) . f1 is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(r,f) . (- f1) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
- h is Relation-like Function-like complex-yielding set
(- 1) * h is Relation-like Function-like complex-yielding set
dom (r,f) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tunit_circle (r + 1))
bool the carrier of (Tunit_circle (r + 1)) is V74() set
x is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
- x is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * x is Relation-like Function-like complex-yielding ext-real-valued real-valued set
f . x is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
y is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
f . y is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(f . x) - (f . y) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(f . x) - (f . y) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (f . y) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (f . y) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(f . y) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . y) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . x) + (- (f . y)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((f . x),(- (f . y))) is Relation-like Function-like set
diffreal .: ((f . x),(f . y)) is Relation-like Function-like set
K254((f . x),(f . y)) is Relation-like Function-like set
K254((f . x),(f . y)) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(f . y) - (f . x) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(f . y) - (f . x) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (f . x) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (f . x) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . x) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . x) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . y) + (- (f . x)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((f . y),(- (f . x))) is Relation-like Function-like set
diffreal .: ((f . y),(f . x)) is Relation-like Function-like set
K254((f . y),(f . x)) is Relation-like Function-like set
K254((f . y),(f . x)) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
- ((f . y) - (f . x)) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
- ((f . y) - (f . x)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(- 1) * ((f . y) - (f . x)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((f . y) - (f . x)) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((f . y) - (f . x)) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
0* r is Relation-like NAT -defined REAL -valued Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of REAL r
REAL r is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered M29( REAL )
K513(r,REAL) is functional non empty FinSequence-membered M29( REAL )
K514(REAL,r,0) is Relation-like empty-yielding NAT -defined REAL -valued Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of K513(r,REAL)
- ((f . x) - (f . y)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * ((f . x) - (f . y)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((f . x) - (f . y)) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((f . x) - (f . y)) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(r,((f . x) - (f . y))) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL r)),1))
the carrier of (Tcircle ((0. (TOP-REAL r)),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
|.((f . x) - (f . y)).| is complex real ext-real non negative Element of REAL
sqr ((f . x) - (f . y)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (((f . x) - (f . y)),((f . x) - (f . y))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (((f . x) - (f . y)),((f . x) - (f . y))) is Relation-like Function-like set
K254(((f . x) - (f . y)),((f . x) - (f . y))) is Relation-like Function-like set
K254(((f . x) - (f . y)),((f . x) - (f . y))) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((f . x) - (f . y)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr ((f . x) - (f . y))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((f . x) - (f . y)))) is complex real ext-real Element of REAL
((f . x) - (f . y)) (/) |.((f . x) - (f . y)).| is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
1 / |.((f . x) - (f . y)).| is complex real ext-real non negative set
|.((f . x) - (f . y)).| " is complex real ext-real non negative set
1 * (|.((f . x) - (f . y)).| ") is complex real ext-real non negative set
(1 / |.((f . x) - (f . y)).|) * ((f . x) - (f . y)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 / |.((f . x) - (f . y)).|) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((1 / |.((f . x) - (f . y)).|),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (1 / |.((f . x) - (f . y)).|) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 / |.((f . x) - (f . y)).|)} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 / |.((f . x) - (f . y)).|)}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 / |.((f . x) - (f . y)).|)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 / |.((f . x) - (f . y)).|)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((f . x) - (f . y)) (#) ((1 / |.((f . x) - (f . y)).|) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|.(- ((f . y) - (f . x))).| is complex real ext-real non negative Element of REAL
sqr (- ((f . y) - (f . x))) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((- ((f . y) - (f . x))),(- ((f . y) - (f . x)))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((- ((f . y) - (f . x))),(- ((f . y) - (f . x)))) is Relation-like Function-like set
K254((- ((f . y) - (f . x))),(- ((f . y) - (f . x)))) is Relation-like Function-like set
K254((- ((f . y) - (f . x))),(- ((f . y) - (f . x)))) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(- ((f . y) - (f . x))) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (- ((f . y) - (f . x)))) is complex real ext-real Element of REAL
sqrt (Sum (sqr (- ((f . y) - (f . x))))) is complex real ext-real Element of REAL
((f . x) - (f . y)) (/) |.(- ((f . y) - (f . x))).| is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
1 / |.(- ((f . y) - (f . x))).| is complex real ext-real non negative set
|.(- ((f . y) - (f . x))).| " is complex real ext-real non negative set
1 * (|.(- ((f . y) - (f . x))).| ") is complex real ext-real non negative set
(1 / |.(- ((f . y) - (f . x))).|) * ((f . x) - (f . y)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 / |.(- ((f . y) - (f . x))).|) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((1 / |.(- ((f . y) - (f . x))).|),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (1 / |.(- ((f . y) - (f . x))).|) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 / |.(- ((f . y) - (f . x))).|)} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 / |.(- ((f . y) - (f . x))).|)}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 / |.(- ((f . y) - (f . x))).|)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 / |.(- ((f . y) - (f . x))).|)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((f . x) - (f . y)) (#) ((1 / |.(- ((f . y) - (f . x))).|) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- ((f . y) - (f . x))) (/) |.(- ((f . y) - (f . x))).| is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(1 / |.(- ((f . y) - (f . x))).|) * (- ((f . y) - (f . x))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- ((f . y) - (f . x))) (#) ((1 / |.(- ((f . y) - (f . x))).|) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((f . y) - (f . x)) (/) |.(- ((f . y) - (f . x))).| is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 / |.(- ((f . y) - (f . x))).|) * ((f . y) - (f . x)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((f . y) - (f . x)) (#) ((1 / |.(- ((f . y) - (f . x))).|) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- (((f . y) - (f . x)) (/) |.(- ((f . y) - (f . x))).|) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (((f . y) - (f . x)) (/) |.(- ((f . y) - (f . x))).|) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(((f . y) - (f . x)) (/) |.(- ((f . y) - (f . x))).|) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(((f . y) - (f . x)) (/) |.(- ((f . y) - (f . x))).|) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|.((f . y) - (f . x)).| is complex real ext-real non negative Element of REAL
sqr ((f . y) - (f . x)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (((f . y) - (f . x)),((f . y) - (f . x))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (((f . y) - (f . x)),((f . y) - (f . x))) is Relation-like Function-like set
K254(((f . y) - (f . x)),((f . y) - (f . x))) is Relation-like Function-like set
K254(((f . y) - (f . x)),((f . y) - (f . x))) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((f . y) - (f . x)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr ((f . y) - (f . x))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((f . y) - (f . x)))) is complex real ext-real Element of REAL
((f . y) - (f . x)) (/) |.((f . y) - (f . x)).| is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
1 / |.((f . y) - (f . x)).| is complex real ext-real non negative set
|.((f . y) - (f . x)).| " is complex real ext-real non negative set
1 * (|.((f . y) - (f . x)).| ") is complex real ext-real non negative set
(1 / |.((f . y) - (f . x)).|) * ((f . y) - (f . x)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 / |.((f . y) - (f . x)).|) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((1 / |.((f . y) - (f . x)).|),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (1 / |.((f . y) - (f . x)).|) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 / |.((f . y) - (f . x)).|)} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 / |.((f . y) - (f . x)).|)}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 / |.((f . y) - (f . x)).|)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 / |.((f . y) - (f . x)).|)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((f . y) - (f . x)) (#) ((1 / |.((f . y) - (f . x)).|) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- (((f . y) - (f . x)) (/) |.((f . y) - (f . x)).|) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
- (((f . y) - (f . x)) (/) |.((f . y) - (f . x)).|) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(- 1) * (((f . y) - (f . x)) (/) |.((f . y) - (f . x)).|) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(((f . y) - (f . x)) (/) |.((f . y) - (f . x)).|) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(((f . y) - (f . x)) (/) |.((f . y) - (f . x)).|) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(r,((f . y) - (f . x))) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL r)),1))
- (r,((f . y) - (f . x))) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * (r,((f . y) - (f . x))) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
r is non empty complex ordinal natural real integer ext-real positive non negative set
r + 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
TOP-REAL (r + 1) is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
0. (TOP-REAL (r + 1)) is Relation-like NAT -defined Function-like V32() V39(r + 1) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
the carrier of (TOP-REAL (r + 1)) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (r + 1)) is Relation-like NAT -defined Function-like V32() V39(r + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
Tcircle ((0. (TOP-REAL (r + 1))),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
Sphere ((0. (TOP-REAL (r + 1))),1) is functional non empty closed V214( TOP-REAL (r + 1)) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (r + 1))
bool the carrier of (TOP-REAL (r + 1)) is V74() set
(TOP-REAL (r + 1)) | (Sphere ((0. (TOP-REAL (r + 1))),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
TOP-REAL r is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):] is set
r NormF is Relation-like the carrier of (TOP-REAL r) -defined the carrier of R^1 -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL r), the carrier of R^1:]
[: the carrier of (TOP-REAL r), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL r), the carrier of R^1:] is set
p is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):]
p (-) is Relation-like Function-like set
(r,p) is Relation-like the carrier of (Tunit_circle (r + 1)) -defined the carrier of (Tunit_circle r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):]
Tunit_circle (r + 1) is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
the carrier of (Tunit_circle (r + 1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
Tunit_circle r is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
0. (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
the ZeroF of (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
Tcircle ((0. (TOP-REAL r)),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Sphere ((0. (TOP-REAL r)),1) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is V74() set
(TOP-REAL r) | (Sphere ((0. (TOP-REAL r)),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
the carrier of (Tunit_circle r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):] is Relation-like set
bool [: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):] is set
f is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):]
e is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):]
p <--> f is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):]
the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) is set
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
(DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is Relation-like set
bool [:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is set
(r NormF) * e is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of R^1:]
[: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of R^1:] is set
e </> ((r NormF) * e) is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * e)) -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * e))),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
dom ((r NormF) * e) is non empty set
the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * e)) is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * e))),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * e))),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
((r NormF) * e) " is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined Function-like total complex-yielding ext-real-valued real-valued set
e <#> (((r NormF) * e) ") is Relation-like Function-like set
dom e is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) is V74() set
dom (r NormF) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
rng e is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
dom ((r NormF) * e) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
dom (e </> ((r NormF) * e)) is Element of bool ( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * e)))
bool ( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * e))) is set
(dom e) /\ (dom ((r NormF) * e)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
dom (r,p) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tunit_circle (r + 1))
bool the carrier of (Tunit_circle (r + 1)) is V74() set
dom (r,p) is non empty set
hy is set
(r,p) . hy is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(e </> ((r NormF) * e)) . hy is Relation-like Function-like complex-yielding ext-real-valued real-valued set
c₃₂ is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
- c₃₂ is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * c₃₂ is Relation-like Function-like complex-yielding ext-real-valued real-valued set
p . c₃₂ is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
Ch is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
p . Ch is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(p . c₃₂) - (p . Ch) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(p . c₃₂) - (p . Ch) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
- (p . Ch) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (p . Ch) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(p . Ch) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p . Ch) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p . c₃₂) + (- (p . Ch)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((p . c₃₂),(- (p . Ch))) is Relation-like Function-like set
diffreal .: ((p . c₃₂),(p . Ch)) is Relation-like Function-like set
K254((p . c₃₂),(p . Ch)) is Relation-like Function-like set
K254((p . c₃₂),(p . Ch)) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
dom f is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
e . c₃₂ is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
f . c₃₂ is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(p . c₃₂) - (f . c₃₂) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- (f . c₃₂) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (f . c₃₂) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . c₃₂) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . c₃₂) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(p . c₃₂) + (- (f . c₃₂)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((p . c₃₂),(- (f . c₃₂))) is Relation-like Function-like set
K254((p . c₃₂),(- (f . c₃₂))) is Relation-like Function-like set
K254((p . c₃₂),(- (f . c₃₂))) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
diffreal .: ((p . c₃₂),(f . c₃₂)) is Relation-like Function-like set
K254((p . c₃₂),(f . c₃₂)) is Relation-like Function-like set
K254((p . c₃₂),(f . c₃₂)) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((r NormF) * e) . c₃₂ is complex real ext-real Element of the carrier of R^1
(r NormF) . (e . c₃₂) is complex real ext-real Element of the carrier of R^1
|.((p . c₃₂) - (p . Ch)).| is complex real ext-real non negative Element of REAL
sqr ((p . c₃₂) - (p . Ch)) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (((p . c₃₂) - (p . Ch)),((p . c₃₂) - (p . Ch))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (((p . c₃₂) - (p . Ch)),((p . c₃₂) - (p . Ch))) is Relation-like Function-like set
K254(((p . c₃₂) - (p . Ch)),((p . c₃₂) - (p . Ch))) is Relation-like Function-like set
K254(((p . c₃₂) - (p . Ch)),((p . c₃₂) - (p . Ch))) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
((p . c₃₂) - (p . Ch)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr ((p . c₃₂) - (p . Ch))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((p . c₃₂) - (p . Ch)))) is complex real ext-real Element of REAL
(r,p) . c₃₂ is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(r,((p . c₃₂) - (p . Ch))) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL r)),1))
the carrier of (Tcircle ((0. (TOP-REAL r)),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
(e . c₃₂) (/) (((r NormF) * e) . c₃₂) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
1 / (((r NormF) * e) . c₃₂) is complex real ext-real set
(((r NormF) * e) . c₃₂) " is complex real ext-real set
1 * ((((r NormF) * e) . c₃₂) ") is complex real ext-real set
(1 / (((r NormF) * e) . c₃₂)) * (e . c₃₂) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(1 / (((r NormF) * e) . c₃₂)) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((1 / (((r NormF) * e) . c₃₂)),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (1 / (((r NormF) * e) . c₃₂)) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(1 / (((r NormF) * e) . c₃₂))} is non empty V195() V196() V197() set
[:(dom (id REAL)),{(1 / (((r NormF) * e) . c₃₂))}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (1 / (((r NormF) * e) . c₃₂))),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (1 / (((r NormF) * e) . c₃₂))),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(e . c₃₂) (#) ((1 / (((r NormF) * e) . c₃₂)) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(((r NormF) * e) ") . c₃₂ is complex real ext-real set
(e . c₃₂) (#) ((((r NormF) * e) ") . c₃₂) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
((((r NormF) * e) ") . c₃₂) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] (((((r NormF) * e) ") . c₃₂),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> ((((r NormF) * e) ") . c₃₂) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{((((r NormF) * e) ") . c₃₂)} is non empty V195() V196() V197() set
[:(dom (id REAL)),{((((r NormF) * e) ") . c₃₂)}:] is Relation-like complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> ((((r NormF) * e) ") . c₃₂)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> ((((r NormF) * e) ") . c₃₂)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(e . c₃₂) (#) (((((r NormF) * e) ") . c₃₂) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(e </> ((r NormF) * e)) . c₃₂ is Relation-like Function-like complex-yielding ext-real-valued real-valued set
r is non empty complex ordinal natural real integer ext-real positive non negative set
r + 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
TOP-REAL (r + 1) is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
0. (TOP-REAL (r + 1)) is Relation-like NAT -defined Function-like V32() V39(r + 1) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
the carrier of (TOP-REAL (r + 1)) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (r + 1)) is Relation-like NAT -defined Function-like V32() V39(r + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
Tcircle ((0. (TOP-REAL (r + 1))),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
Sphere ((0. (TOP-REAL (r + 1))),1) is functional non empty closed V214( TOP-REAL (r + 1)) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (r + 1))
bool the carrier of (TOP-REAL (r + 1)) is V74() set
(TOP-REAL (r + 1)) | (Sphere ((0. (TOP-REAL (r + 1))),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
TOP-REAL r is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):] is set
Tunit_circle (r + 1) is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
Tunit_circle r is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
0. (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
the ZeroF of (TOP-REAL r) is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
Tcircle ((0. (TOP-REAL r)),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
Sphere ((0. (TOP-REAL r)),1) is functional non empty closed complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is V74() set
(TOP-REAL r) | (Sphere ((0. (TOP-REAL r)),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL r
f is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):]
(r,f) is Relation-like the carrier of (Tunit_circle (r + 1)) -defined the carrier of (Tunit_circle r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):]
the carrier of (Tunit_circle (r + 1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
the carrier of (Tunit_circle r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):] is Relation-like set
bool [: the carrier of (Tunit_circle (r + 1)), the carrier of (Tunit_circle r):] is set
f (-) is Relation-like Function-like set
f1 is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):]
f <--> f1 is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):]
the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) is set
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
(DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is Relation-like set
bool [:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is set
h is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of (TOP-REAL r):]
r NormF is Relation-like the carrier of (TOP-REAL r) -defined the carrier of R^1 -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL r), the carrier of R^1:]
[: the carrier of (TOP-REAL r), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL r), the carrier of R^1:] is set
(r NormF) * h is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of R^1:]
[: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)), the carrier of R^1:] is set
dom f1 is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) is V74() set
dom h is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
y is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
((r NormF) * h) . y is complex real ext-real Element of the carrier of R^1
h . y is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(r NormF) . (h . y) is complex real ext-real Element of the carrier of R^1
|.(h . y).| is complex real ext-real non negative Element of REAL
sqr (h . y) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((h . y),(h . y)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((h . y),(h . y)) is Relation-like Function-like set
K254((h . y),(h . y)) is Relation-like Function-like set
K254((h . y),(h . y)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(h . y) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (h . y)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (h . y))) is complex real ext-real Element of REAL
y is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
h . y is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
f . y is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
f1 . y is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
(f . y) - (f1 . y) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- (f1 . y) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) * (f1 . y) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(f1 . y) (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f1 . y) (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(f . y) + (- (f1 . y)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K143() .: ((f . y),(- (f1 . y))) is Relation-like Function-like set
K254((f . y),(- (f1 . y))) is Relation-like Function-like set
K254((f . y),(- (f1 . y))) (#) K143() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
diffreal .: ((f . y),(f1 . y)) is Relation-like Function-like set
K254((f . y),(f1 . y)) is Relation-like Function-like set
K254((f . y),(f1 . y)) (#) diffreal is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
- y is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * y is Relation-like Function-like complex-yielding ext-real-valued real-valued set
f . (- y) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(f . y) - (f . (- y)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
- (f . (- y)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * (f . (- y)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(f . y) + (- (f . (- y))) is Relation-like NAT -defined Function-like complex-yielding ext-real-valued real-valued set
|.(h . y).| is complex real ext-real non negative Element of REAL
sqr (h . y) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((h . y),(h . y)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((h . y),(h . y)) is Relation-like Function-like set
K254((h . y),(h . y)) is Relation-like Function-like set
K254((h . y),(h . y)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(h . y) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (h . y)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (h . y))) is complex real ext-real Element of REAL
y is set
dom ((r NormF) * h) is non empty set
((r NormF) * h) . y is complex real ext-real set
dom ((r NormF) * h) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
hx is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1))
((r NormF) * h) . hx is complex real ext-real Element of the carrier of R^1
h . hx is Relation-like NAT -defined Function-like V32() V39(r) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL r)
|.(h . hx).| is complex real ext-real non negative Element of REAL
sqr (h . hx) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt ((h . hx),(h . hx)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: ((h . hx),(h . hx)) is Relation-like Function-like set
K254((h . hx),(h . hx)) is Relation-like Function-like set
K254((h . hx),(h . hx)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
(h . hx) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr (h . hx)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (h . hx))) is complex real ext-real Element of REAL
h </> ((r NormF) * h) is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * h)) -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * h))),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
dom ((r NormF) * h) is non empty set
the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * h)) is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * h))),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:( the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) /\ (dom ((r NormF) * h))),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
((r NormF) * h) " is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (r + 1))),1)) -defined Function-like total complex-yielding ext-real-valued real-valued set
h <#> (((r NormF) * h) ") is Relation-like Function-like set
the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)), the carrier of (TOP-REAL 2):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)), the carrier of (TOP-REAL 2):] is set
2 + 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
Tunit_circle (2 + 1) is non empty TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (2 + 1)
TOP-REAL (2 + 1) is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
0. (TOP-REAL (2 + 1)) is Relation-like NAT -defined Function-like V32() V39(2 + 1) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (2 + 1))
the carrier of (TOP-REAL (2 + 1)) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (2 + 1)) is Relation-like NAT -defined Function-like V32() V39(2 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (2 + 1))
Tcircle ((0. (TOP-REAL (2 + 1))),1) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (2 + 1)
Sphere ((0. (TOP-REAL (2 + 1))),1) is functional non empty closed V214( TOP-REAL (2 + 1)) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (2 + 1))
bool the carrier of (TOP-REAL (2 + 1)) is V74() set
(TOP-REAL (2 + 1)) | (Sphere ((0. (TOP-REAL (2 + 1))),1)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (2 + 1)
the carrier of (Tunit_circle (2 + 1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
(1) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 3) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued nullhomotopic Path of (),()
r is complex real ext-real set
r + (1 / 2) is complex real ext-real set
p is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)), the carrier of (TOP-REAL 2):]
(2,p) is Relation-like the carrier of (Tunit_circle (2 + 1)) -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):]
[: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):] is set
(2,p) * (1) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
H₁(p) . (r + (1 / 2)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
H₁(p) . r is Relation-like Function-like complex-yielding ext-real-valued real-valued set
- (H₁(p) . r) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * (H₁(p) . r) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
dom (2,p) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tunit_circle (2 + 1))
bool the carrier of (Tunit_circle (2 + 1)) is V74() set
h is complex real ext-real Element of the carrier of I[01]
(1) . h is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 3)
- ((1) . h) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * ((1) . h) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
the carrier of (Tcircle ((0. (TOP-REAL 3)),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
(1) . (r + (1 / 2)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
((2 * PI) * 1) * (r + (1 / 2)) is complex real ext-real Element of REAL
cos (((2 * PI) * 1) * (r + (1 / 2))) is complex real ext-real Element of REAL
sin (((2 * PI) * 1) * (r + (1 / 2))) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * 1) * (r + (1 / 2)))),(sin (((2 * PI) * 1) * (r + (1 / 2)))),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos (((2 * PI) * 1) * (r + (1 / 2))))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos (((2 * PI) * 1) * (r + (1 / 2))))] is set
{1,(cos (((2 * PI) * 1) * (r + (1 / 2))))} is non empty V195() V196() V197() set
{{1,(cos (((2 * PI) * 1) * (r + (1 / 2))))},{1}} is non empty set
{[1,(cos (((2 * PI) * 1) * (r + (1 / 2))))]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * 1) * (r + (1 / 2))))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * 1) * (r + (1 / 2))))] is set
{1,(sin (((2 * PI) * 1) * (r + (1 / 2))))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * 1) * (r + (1 / 2))))},{1}} is non empty set
{[1,(sin (((2 * PI) * 1) * (r + (1 / 2))))]} is Relation-like Function-like non empty set
<*(cos (((2 * PI) * 1) * (r + (1 / 2))))*> ^ <*(sin (((2 * PI) * 1) * (r + (1 / 2))))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos (((2 * PI) * 1) * (r + (1 / 2))))*> ^ <*(sin (((2 * PI) * 1) * (r + (1 / 2))))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(2 * PI) * r is complex real ext-real Element of REAL
cos ((2 * PI) * r) is complex real ext-real Element of REAL
- (cos ((2 * PI) * r)) is complex real ext-real Element of REAL
((2 * PI) * r) + PI is complex real ext-real Element of REAL
sin (((2 * PI) * r) + PI) is complex real ext-real Element of REAL
|[(- (cos ((2 * PI) * r))),(sin (((2 * PI) * r) + PI)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(- (cos ((2 * PI) * r)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(- (cos ((2 * PI) * r)))] is set
{1,(- (cos ((2 * PI) * r)))} is non empty V195() V196() V197() set
{{1,(- (cos ((2 * PI) * r)))},{1}} is non empty set
{[1,(- (cos ((2 * PI) * r)))]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * r) + PI))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * r) + PI))] is set
{1,(sin (((2 * PI) * r) + PI))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * r) + PI))},{1}} is non empty set
{[1,(sin (((2 * PI) * r) + PI))]} is Relation-like Function-like non empty set
<*(- (cos ((2 * PI) * r)))*> ^ <*(sin (((2 * PI) * r) + PI))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(- (cos ((2 * PI) * r)))*> ^ <*(sin (((2 * PI) * r) + PI))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
sin ((2 * PI) * r) is complex real ext-real Element of REAL
- (sin ((2 * PI) * r)) is complex real ext-real Element of REAL
- Q is complex real integer ext-real non positive set
|[(- (cos ((2 * PI) * r))),(- (sin ((2 * PI) * r))),(- Q)]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(- (sin ((2 * PI) * r)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(- (sin ((2 * PI) * r)))] is set
{1,(- (sin ((2 * PI) * r)))} is non empty V195() V196() V197() set
{{1,(- (sin ((2 * PI) * r)))},{1}} is non empty set
{[1,(- (sin ((2 * PI) * r)))]} is Relation-like Function-like non empty set
<*(- (cos ((2 * PI) * r)))*> ^ <*(- (sin ((2 * PI) * r)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
<*(- Q)*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(- Q)] is set
{1,(- Q)} is non empty V195() V196() V197() V198() V199() set
{{1,(- Q)},{1}} is non empty set
{[1,(- Q)]} is Relation-like Function-like non empty set
(<*(- (cos ((2 * PI) * r)))*> ^ <*(- (sin ((2 * PI) * r)))*>) ^ <*(- Q)*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
((2 * PI) * 1) * r is complex real ext-real Element of REAL
cos (((2 * PI) * 1) * r) is complex real ext-real Element of REAL
sin (((2 * PI) * 1) * r) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * 1) * r)),(sin (((2 * PI) * 1) * r)),0]| is Relation-like NAT -defined Function-like non empty V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
<*(cos (((2 * PI) * 1) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos (((2 * PI) * 1) * r))] is set
{1,(cos (((2 * PI) * 1) * r))} is non empty V195() V196() V197() set
{{1,(cos (((2 * PI) * 1) * r))},{1}} is non empty set
{[1,(cos (((2 * PI) * 1) * r))]} is Relation-like Function-like non empty set
<*(sin (((2 * PI) * 1) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin (((2 * PI) * 1) * r))] is set
{1,(sin (((2 * PI) * 1) * r))} is non empty V195() V196() V197() set
{{1,(sin (((2 * PI) * 1) * r))},{1}} is non empty set
{[1,(sin (((2 * PI) * 1) * r))]} is Relation-like Function-like non empty set
<*(cos (((2 * PI) * 1) * r))*> ^ <*(sin (((2 * PI) * 1) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(<*(cos (((2 * PI) * 1) * r))*> ^ <*(sin (((2 * PI) * 1) * r))*>) ^ <*0*> is Relation-like NAT -defined Function-like non empty V32() V39((1 + 1) + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- |[(cos (((2 * PI) * 1) * r)),(sin (((2 * PI) * 1) * r)),0]| is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
- |[(cos (((2 * PI) * 1) * r)),(sin (((2 * PI) * 1) * r)),0]| is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(- 1) * |[(cos (((2 * PI) * 1) * r)),(sin (((2 * PI) * 1) * r)),0]| is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|[(cos (((2 * PI) * 1) * r)),(sin (((2 * PI) * 1) * r)),0]| (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[(cos (((2 * PI) * 1) * r)),(sin (((2 * PI) * 1) * r)),0]| (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(2,p) . ((1) . (r + (1 / 2))) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(2,p) . ((1) . h) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
- ((2,p) . ((1) . h)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * ((2,p) . ((1) . h)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
H₁(p) . h is Relation-like Function-like complex-yielding ext-real-valued real-valued set
- (H₁(p) . h) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * (H₁(p) . h) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
r is complex real ext-real Element of the carrier of R^1
CircleMap . r is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
p is complex real ext-real Element of the carrier of R^1
CircleMap . p is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
- (CircleMap . p) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * (CircleMap . p) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(2 * PI) * r is complex real ext-real Element of REAL
cos ((2 * PI) * r) is complex real ext-real Element of REAL
sin ((2 * PI) * r) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * r)),(sin ((2 * PI) * r))]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(cos ((2 * PI) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos ((2 * PI) * r))] is set
{1,(cos ((2 * PI) * r))} is non empty V195() V196() V197() set
{{1,(cos ((2 * PI) * r))},{1}} is non empty set
{[1,(cos ((2 * PI) * r))]} is Relation-like Function-like non empty set
<*(sin ((2 * PI) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin ((2 * PI) * r))] is set
{1,(sin ((2 * PI) * r))} is non empty V195() V196() V197() set
{{1,(sin ((2 * PI) * r))},{1}} is non empty set
{[1,(sin ((2 * PI) * r))]} is Relation-like Function-like non empty set
<*(cos ((2 * PI) * r))*> ^ <*(sin ((2 * PI) * r))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(2 * PI) * p is complex real ext-real Element of REAL
cos ((2 * PI) * p) is complex real ext-real Element of REAL
sin ((2 * PI) * p) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * p)),(sin ((2 * PI) * p))]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(cos ((2 * PI) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(cos ((2 * PI) * p))] is set
{1,(cos ((2 * PI) * p))} is non empty V195() V196() V197() set
{{1,(cos ((2 * PI) * p))},{1}} is non empty set
{[1,(cos ((2 * PI) * p))]} is Relation-like Function-like non empty set
<*(sin ((2 * PI) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(sin ((2 * PI) * p))] is set
{1,(sin ((2 * PI) * p))} is non empty V195() V196() V197() set
{{1,(sin ((2 * PI) * p))},{1}} is non empty set
{[1,(sin ((2 * PI) * p))]} is Relation-like Function-like non empty set
<*(cos ((2 * PI) * p))*> ^ <*(sin ((2 * PI) * p))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- |[(cos ((2 * PI) * p)),(sin ((2 * PI) * p))]| is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
- |[(cos ((2 * PI) * p)),(sin ((2 * PI) * p))]| is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
(- 1) * |[(cos ((2 * PI) * p)),(sin ((2 * PI) * p))]| is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
K145() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined INT -valued Function-like constant non empty total complex-yielding ext-real-valued real-valued set
{(- 1)} is non empty V195() V196() V197() V198() V199() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) is Relation-like Function-like set
K254(((dom (id REAL)) --> (- 1)),(id REAL)) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
|[(cos ((2 * PI) * p)),(sin ((2 * PI) * p))]| (#) ((- 1) multreal) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
|[(cos ((2 * PI) * p)),(sin ((2 * PI) * p))]| (#) K141() is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
- (cos ((2 * PI) * p)) is complex real ext-real Element of REAL
- (sin ((2 * PI) * p)) is complex real ext-real Element of REAL
|[(- (cos ((2 * PI) * p))),(- (sin ((2 * PI) * p)))]| is Relation-like NAT -defined Function-like non empty V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
<*(- (cos ((2 * PI) * p)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(- (cos ((2 * PI) * p)))] is set
{1,(- (cos ((2 * PI) * p)))} is non empty V195() V196() V197() set
{{1,(- (cos ((2 * PI) * p)))},{1}} is non empty set
{[1,(- (cos ((2 * PI) * p)))]} is Relation-like Function-like non empty set
<*(- (sin ((2 * PI) * p)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
[1,(- (sin ((2 * PI) * p)))] is set
{1,(- (sin ((2 * PI) * p)))} is non empty V195() V196() V197() set
{{1,(- (sin ((2 * PI) * p)))},{1}} is non empty set
{[1,(- (sin ((2 * PI) * p)))]} is Relation-like Function-like non empty set
<*(- (cos ((2 * PI) * p)))*> ^ <*(- (sin ((2 * PI) * p)))*> is Relation-like NAT -defined Function-like non empty V32() V39(1 + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
PI + ((2 * PI) * p) is complex real ext-real Element of REAL
cos (PI + ((2 * PI) * p)) is complex real ext-real Element of REAL
sin (PI + ((2 * PI) * p)) is complex real ext-real Element of REAL
f is complex real integer ext-real set
(2 * PI) * f is complex real ext-real Element of REAL
(PI + ((2 * PI) * p)) + ((2 * PI) * f) is complex real ext-real Element of REAL
2 * r is complex real ext-real Element of REAL
PI * (2 * r) is complex real ext-real Element of REAL
(PI * (2 * r)) / PI is complex real ext-real Element of COMPLEX
PI " is non empty complex real ext-real positive non negative set
(PI * (2 * r)) * (PI ") is complex real ext-real set
2 * p is complex real ext-real Element of REAL
1 + (2 * p) is complex real ext-real Element of REAL
2 * f is complex real integer ext-real even Element of REAL
(1 + (2 * p)) + (2 * f) is complex real ext-real Element of REAL
PI * ((1 + (2 * p)) + (2 * f)) is complex real ext-real Element of REAL
(PI * ((1 + (2 * p)) + (2 * f))) / PI is complex real ext-real Element of COMPLEX
(PI * ((1 + (2 * p)) + (2 * f))) * (PI ") is complex real ext-real set
(2 * f) + 1 is non empty complex real integer ext-real non even Element of REAL
e is non empty complex real integer ext-real non even Element of REAL
e / 2 is non empty complex real ext-real Element of COMPLEX
e * (2 ") is non empty complex real ext-real set
(p,(e / 2)) is complex real ext-real Element of the carrier of R^1
c100a is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle (2 + 1))
r is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)), the carrier of (TOP-REAL 2):]
(2,r) is Relation-like the carrier of (Tunit_circle (2 + 1)) -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):]
[: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):] is set
(2,r) * (1) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
(2,r) . c100a is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
[: the carrier of (Tunit_circle 3), the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of (Tunit_circle 3), the carrier of (Tunit_circle 2):] is set
p is Relation-like the carrier of (Tunit_circle 3) -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total continuous Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle 3), the carrier of (Tunit_circle 2):]
p * (1) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued nullhomotopic Path of p . (),p . ()
p . () is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
r is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)), the carrier of (TOP-REAL 2):]
f is set
(2,r) is Relation-like the carrier of (Tunit_circle (2 + 1)) -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):]
[: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):] is set
(2,r) . () is Relation-like Function-like complex-yielding ext-real-valued real-valued set
{((2,r) . ())} is functional non empty complex-functions-membered set
CircleMap " {((2,r) . ())} is V195() V196() V197() Element of bool the carrier of K708()
p is complex real ext-real set
(2,r) . c100a is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(2,r) * (1) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
p + (1 / 2) is complex real ext-real set
f1 is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of (2,r) . c100a,(2,r) . c100a
(((2,r) . c100a),((2,r) . c100a),f,f1) is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
e is complex real ext-real Element of the carrier of I[01]
e + (1 / 2) is complex real ext-real set
R^1 (e + (1 / 2)) is complex real ext-real Element of the carrier of K708()
CircleMap * (((2,r) . c100a),((2,r) . c100a),f,f1) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
[: the carrier of I[01], the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of I[01], the carrier of (Tunit_circle 2):] is set
y is complex real ext-real Element of the carrier of I[01]
(CircleMap * (((2,r) . c100a),((2,r) . c100a),f,f1)) . y is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(((2,r) . c100a),((2,r) . c100a),f,f1) . y is complex real ext-real Element of the carrier of R^1
CircleMap . ((((2,r) . c100a),((2,r) . c100a),f,f1) . y) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(CircleMap * (((2,r) . c100a),((2,r) . c100a),f,f1)) . e is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(((2,r) . c100a),((2,r) . c100a),f,f1) . e is complex real ext-real Element of the carrier of R^1
CircleMap . ((((2,r) . c100a),((2,r) . c100a),f,f1) . e) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(((2,r) . c100a),((2,r) . c100a),f,f1) . (R^1 (e + (1 / 2))) is complex real ext-real set
CircleMap . ((((2,r) . c100a),((2,r) . c100a),f,f1) . (R^1 (e + (1 / 2)))) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
- (CircleMap . ((((2,r) . c100a),((2,r) . c100a),f,f1) . e)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(- 1) * (CircleMap . ((((2,r) . c100a),((2,r) . c100a),f,f1) . e)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
hx is non empty complex real integer ext-real non even set
hx / 2 is non empty complex real ext-real Element of COMPLEX
hx * (2 ") is non empty complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),f,f1) . e),(hx / 2)) is complex real ext-real Element of the carrier of R^1
hy is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of (2,r) . c100a,(2,r) . c100a
(((2,r) . c100a),((2,r) . c100a),f,hy) is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
(((2,r) . c100a),((2,r) . c100a),f,hy) . (p + (1 / 2)) is complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),f,hy) . p is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),f,hy) . p) + (hx / 2) is complex real ext-real set
r is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)), the carrier of (TOP-REAL 2):]
p is set
(2,r) is Relation-like the carrier of (Tunit_circle (2 + 1)) -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):]
(2,r) . () is Relation-like Function-like complex-yielding ext-real-valued real-valued set
{((2,r) . ())} is functional non empty complex-functions-membered set
CircleMap " {((2,r) . ())} is V195() V196() V197() Element of bool the carrier of K708()
(2,r) . c100a is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(2,r) * (1) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
f is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of (2,r) . c100a,(2,r) . c100a
(((2,r) . c100a),((2,r) . c100a),p,f) is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
[.0,(1 / 2).] is V195() V196() V197() closed Element of bool REAL
AffineMap (1,(1 / 2)) is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total quasi_total onto bijective complex-yielding ext-real-valued real-valued continuous Element of bool [:REAL,REAL:]
(AffineMap (1,(1 / 2))) | [.0,(1 / 2).] is Relation-like REAL -defined [.0,(1 / 2).] -defined REAL -defined REAL -valued Function-like complex-yielding ext-real-valued real-valued continuous Element of bool [:REAL,REAL:]
(((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]) is Relation-like REAL -defined the carrier of R^1 -valued Function-like complex-yielding ext-real-valued real-valued Element of bool [:REAL, the carrier of R^1:]
[:REAL, the carrier of R^1:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [:REAL, the carrier of R^1:] is set
((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f) is Relation-like the carrier of I[01] -defined Function-like complex-yielding ext-real-valued real-valued set
- (((2,r) . c100a),((2,r) . c100a),p,f) is Relation-like the carrier of I[01] -defined Function-like total complex-yielding ext-real-valued real-valued set
(- 1) * (((2,r) . c100a),((2,r) . c100a),p,f) is Relation-like the carrier of I[01] -defined Function-like total complex-yielding ext-real-valued real-valued set
((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) + (- (((2,r) . c100a),((2,r) . c100a),p,f)) is Relation-like the carrier of I[01] -defined Function-like complex-yielding ext-real-valued real-valued set
2 * (((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f)) is Relation-like the carrier of I[01] -defined Function-like complex-yielding ext-real-valued real-valued set
dom (AffineMap (1,(1 / 2))) is non empty V195() V196() V197() Element of bool REAL
dom ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]) is V195() V196() V197() Element of bool REAL
dom (((2,r) . c100a),((2,r) . c100a),p,f) is non empty V195() V196() V197() Element of bool the carrier of I[01]
bool the carrier of I[01] is V74() set
rng ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]) is V195() V196() V197() Element of bool REAL
y is set
hx is set
((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]) . hx is complex real ext-real set
hy is complex real ext-real set
((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]) . hy is complex real ext-real set
(AffineMap (1,(1 / 2))) . hy is complex real ext-real set
1 * hy is complex real ext-real Element of REAL
(1 * hy) + (1 / 2) is complex real ext-real Element of REAL
hy + (1 / 2) is complex real ext-real set
dom (((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f)) is V195() V196() V197() Element of bool the carrier of I[01]
dom ((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) is V195() V196() V197() Element of bool REAL
(dom ((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]))) /\ (dom (((2,r) . c100a),((2,r) . c100a),p,f)) is V195() V196() V197() Element of bool the carrier of I[01]
(dom ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) /\ (dom (((2,r) . c100a),((2,r) . c100a),p,f)) is V195() V196() V197() Element of bool the carrier of I[01]
dom (2 * (((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f))) is V195() V196() V197() Element of bool the carrier of I[01]
rng (2 * (((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f))) is V195() V196() V197() set
j0 is complex real ext-real Element of the carrier of I[01]
j0 + (1 / 2) is complex real ext-real set
hx is non empty complex real integer ext-real non even set
hx / 2 is non empty complex real ext-real Element of COMPLEX
hx * (2 ") is non empty complex real ext-real set
hy is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of (2,r) . c100a,(2,r) . c100a
(((2,r) . c100a),((2,r) . c100a),p,hy) is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
c₃₂ is complex real ext-real set
c₃₂ + (1 / 2) is complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),p,hy) . (c₃₂ + (1 / 2)) is complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),p,hy) . c₃₂ is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),p,hy) . c₃₂) + (hx / 2) is complex real ext-real set
((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]) . 0 is complex real ext-real set
(AffineMap (1,(1 / 2))) . 0 is complex real ext-real Element of REAL
1 * Q is complex ordinal natural real integer non irrational ext-real non negative V195() V196() V197() V198() V199() V200() Element of NAT
(1 * Q) + (1 / 2) is non empty complex real ext-real positive non negative Element of REAL
((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]) . c₃₂ is complex real ext-real set
(AffineMap (1,(1 / 2))) . c₃₂ is complex real ext-real set
1 * c₃₂ is complex real ext-real Element of REAL
(1 * c₃₂) + (1 / 2) is complex real ext-real Element of REAL
X is non empty complex real integer ext-real non even set
X / 2 is non empty complex real ext-real Element of COMPLEX
X * (2 ") is non empty complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),p,f) . (j0 + (1 / 2)) is complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),p,f) . j0 is complex real ext-real Element of the carrier of R^1
(((((2,r) . c100a),((2,r) . c100a),p,f) . j0),(hx / 2)) is complex real ext-real Element of the carrier of R^1
(((2,r) . c100a),((2,r) . c100a),p,f) . (c₃₂ + (1 / 2)) is complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),p,f) . c₃₂ is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),p,f) . c₃₂) + (X / 2) is complex real ext-real set
y is Relation-like REAL -defined REAL -valued Function-like complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
y . 0 is complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f)) . 0 is complex real ext-real set
2 * ((((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f)) . 0) is complex real ext-real Element of REAL
((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . 0 is complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),p,f) . 0 is complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . 0) - ((((2,r) . c100a),((2,r) . c100a),p,f) . 0) is complex real ext-real set
- ((((2,r) . c100a),((2,r) . c100a),p,f) . 0) is complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . 0) + (- ((((2,r) . c100a),((2,r) . c100a),p,f) . 0)) is complex real ext-real set
2 * ((((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . 0) - ((((2,r) . c100a),((2,r) . c100a),p,f) . 0)) is complex real ext-real Element of REAL
(((2,r) . c100a),((2,r) . c100a),p,f) . (1 / 2) is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),p,f) . (1 / 2)) - ((((2,r) . c100a),((2,r) . c100a),p,f) . 0) is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),p,f) . (1 / 2)) + (- ((((2,r) . c100a),((2,r) . c100a),p,f) . 0)) is complex real ext-real set
2 * (((((2,r) . c100a),((2,r) . c100a),p,f) . (1 / 2)) - ((((2,r) . c100a),((2,r) . c100a),p,f) . 0)) is complex real ext-real Element of REAL
2 * (hx / 2) is non empty complex real ext-real Element of REAL
y . c₃₂ is complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f)) . c₃₂ is complex real ext-real set
2 * ((((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f)) . c₃₂) is complex real ext-real Element of REAL
((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . c₃₂ is complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . c₃₂) - ((((2,r) . c100a),((2,r) . c100a),p,f) . c₃₂) is complex real ext-real set
- ((((2,r) . c100a),((2,r) . c100a),p,f) . c₃₂) is complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . c₃₂) + (- ((((2,r) . c100a),((2,r) . c100a),p,f) . c₃₂)) is complex real ext-real set
2 * ((((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . c₃₂) - ((((2,r) . c100a),((2,r) . c100a),p,f) . c₃₂)) is complex real ext-real Element of REAL
((((2,r) . c100a),((2,r) . c100a),p,f) . (c₃₂ + (1 / 2))) - ((((2,r) . c100a),((2,r) . c100a),p,f) . c₃₂) is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),p,f) . (c₃₂ + (1 / 2))) + (- ((((2,r) . c100a),((2,r) . c100a),p,f) . c₃₂)) is complex real ext-real set
2 * (((((2,r) . c100a),((2,r) . c100a),p,f) . (c₃₂ + (1 / 2))) - ((((2,r) . c100a),((2,r) . c100a),p,f) . c₃₂)) is complex real ext-real Element of REAL
2 * (X / 2) is non empty complex real ext-real Element of REAL
rng y is V195() V196() V197() Element of bool REAL
xx is set
dom y is V195() V196() V197() Element of bool REAL
Cx is set
y . Cx is complex real ext-real set
F is complex real ext-real set
((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]) . F is complex real ext-real set
(AffineMap (1,(1 / 2))) . F is complex real ext-real set
1 * F is complex real ext-real Element of REAL
(1 * F) + (1 / 2) is complex real ext-real Element of REAL
F + (1 / 2) is complex real ext-real set
S is non empty complex real integer ext-real non even set
S / 2 is non empty complex real ext-real Element of COMPLEX
S * (2 ") is non empty complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),p,f) . (F + (1 / 2)) is complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),p,f) . F is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),p,f) . F) + (S / 2) is complex real ext-real set
y . F is complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f)) . F is complex real ext-real set
2 * ((((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - (((2,r) . c100a),((2,r) . c100a),p,f)) . F) is complex real ext-real Element of REAL
((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . F is complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . F) - ((((2,r) . c100a),((2,r) . c100a),p,f) . F) is complex real ext-real set
- ((((2,r) . c100a),((2,r) . c100a),p,f) . F) is complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . F) + (- ((((2,r) . c100a),((2,r) . c100a),p,f) . F)) is complex real ext-real set
2 * ((((((2,r) . c100a),((2,r) . c100a),p,f) * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) . F) - ((((2,r) . c100a),((2,r) . c100a),p,f) . F)) is complex real ext-real Element of REAL
((((2,r) . c100a),((2,r) . c100a),p,f) . (F + (1 / 2))) - ((((2,r) . c100a),((2,r) . c100a),p,f) . F) is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),p,f) . (F + (1 / 2))) + (- ((((2,r) . c100a),((2,r) . c100a),p,f) . F)) is complex real ext-real set
2 * (((((2,r) . c100a),((2,r) . c100a),p,f) . (F + (1 / 2))) - ((((2,r) . c100a),((2,r) . c100a),p,f) . F)) is complex real ext-real Element of REAL
2 * (S / 2) is non empty complex real ext-real Element of REAL
Closed-Interval-TSpace (0,(1 / 2)) is non empty strict TopSpace-like V231() SubSpace of R^1
the carrier of (Closed-Interval-TSpace (0,(1 / 2))) is non empty V195() V196() V197() set
Cx is V195() V196() V197() Element of bool the carrier of R^1
R^1 | Cx is strict TopSpace-like V231() SubSpace of R^1
the carrier of (R^1 | Cx) is V195() V196() V197() set
[: the carrier of (Closed-Interval-TSpace (0,(1 / 2))), the carrier of (R^1 | Cx):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (Closed-Interval-TSpace (0,(1 / 2))), the carrier of (R^1 | Cx):] is set
dom y is V195() V196() V197() Element of bool REAL
R^1 (dom y) is V195() V196() V197() Element of bool the carrier of K708()
R^1 | (R^1 (dom y)) is strict TopSpace-like V231() SubSpace of R^1
[: the carrier of (Closed-Interval-TSpace (0,(1 / 2))), the carrier of I[01]:] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (Closed-Interval-TSpace (0,(1 / 2))), the carrier of I[01]:] is set
R^1 ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]) is Relation-like the carrier of (K708() | (R^1 (dom ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])))) -defined the carrier of (K708() | (R^1 (rng ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])))) -valued Function-like quasi_total onto continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (K708() | (R^1 (dom ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])))), the carrier of (K708() | (R^1 (rng ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])))):]
R^1 (dom ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) is V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (dom ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]))) is strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (dom ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])))) is V195() V196() V197() set
R^1 (rng ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) is V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (rng ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]))) is strict TopSpace-like V231() SubSpace of K708()
the carrier of (K708() | (R^1 (rng ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])))) is V195() V196() V197() set
[: the carrier of (K708() | (R^1 (dom ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])))), the carrier of (K708() | (R^1 (rng ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])))):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (K708() | (R^1 (dom ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])))), the carrier of (K708() | (R^1 (rng ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])))):] is set
rng (((2,r) . c100a),((2,r) . c100a),p,f) is non empty V195() V196() V197() Element of bool the carrier of R^1
A is Relation-like REAL -defined REAL -valued Function-like complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
A * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).]) is Relation-like REAL -defined REAL -valued Function-like complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
(A * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) - A is Relation-like REAL -defined REAL -valued Function-like complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
- A is Relation-like REAL -defined Function-like complex-yielding ext-real-valued real-valued set
(- 1) * A is Relation-like REAL -defined Function-like complex-yielding ext-real-valued real-valued set
(A * ((AffineMap (1,(1 / 2))) | [.0,(1 / 2).])) + (- A) is Relation-like REAL -defined Function-like complex-yielding ext-real-valued real-valued set
H is Relation-like REAL -defined REAL -valued Function-like complex-yielding ext-real-valued real-valued continuous Element of bool [:REAL,REAL:]
rng H is V195() V196() V197() Element of bool REAL
R^1 (rng H) is V195() V196() V197() Element of bool the carrier of K708()
R^1 | (R^1 (rng H)) is strict TopSpace-like V231() SubSpace of R^1
the carrier of (R^1 | (R^1 (rng H))) is V195() V196() V197() set
dom H is V195() V196() V197() Element of bool REAL
R^1 (dom H) is V195() V196() V197() Element of bool the carrier of K708()
K708() | (R^1 (dom H)) is strict TopSpace-like V231() SubSpace of K708()
K708() | (R^1 (rng H)) is strict TopSpace-like V231() SubSpace of K708()
R^1 H is Relation-like the carrier of (K708() | (R^1 (dom H))) -defined the carrier of (K708() | (R^1 (rng H))) -valued Function-like quasi_total onto continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (K708() | (R^1 (dom H))), the carrier of (K708() | (R^1 (rng H))):]
the carrier of (K708() | (R^1 (dom H))) is V195() V196() V197() set
the carrier of (K708() | (R^1 (rng H))) is V195() V196() V197() set
[: the carrier of (K708() | (R^1 (dom H))), the carrier of (K708() | (R^1 (rng H))):] is Relation-like complex-yielding ext-real-valued real-valued set
bool [: the carrier of (K708() | (R^1 (dom H))), the carrier of (K708() | (R^1 (rng H))):] is set
F is Relation-like the carrier of (Closed-Interval-TSpace (0,(1 / 2))) -defined the carrier of (R^1 | Cx) -valued Function-like quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of (Closed-Interval-TSpace (0,(1 / 2))), the carrier of (R^1 | Cx):]
r is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)), the carrier of (TOP-REAL 2):]
(2,r) is Relation-like the carrier of (Tunit_circle (2 + 1)) -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):]
(2,r) . c100a is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(2,r) * (1) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
p is complex real ext-real Element of the carrier of R^1
f is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of (2,r) . c100a,(2,r) . c100a
(((2,r) . c100a),((2,r) . c100a),p,f) is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
(((2,r) . c100a),((2,r) . c100a),p,f) . 1 is complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),p,f) . 0 is complex real ext-real set
e is complex real integer ext-real set
e / 2 is complex real ext-real Element of COMPLEX
e * (2 ") is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),p,f) . 0) + e is complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),p,f) . (Q + (1 / 2)) is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),p,f) . (Q + (1 / 2))) + (e / 2) is complex real ext-real set
((((2,r) . c100a),((2,r) . c100a),p,f) . 0) + (e / 2) is complex real ext-real set
(((((2,r) . c100a),((2,r) . c100a),p,f) . 0) + (e / 2)) + (e / 2) is complex real ext-real set
r is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)), the carrier of (TOP-REAL 2):]
(2,r) is Relation-like the carrier of (Tunit_circle (2 + 1)) -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):]
(2,r) * (1) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
(2,r) . c100a is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
p is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of (2,r) . c100a,(2,r) . c100a
f1 is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like constant non empty total quasi_total continuous Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued nullhomotopic Path of (2,r) . c100a,(2,r) . c100a
rng CircleMap is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tunit_circle 2)
bool the carrier of (Tunit_circle 2) is V74() set
h is set
CircleMap . h is Relation-like Function-like complex-yielding ext-real-valued real-valued set
{((2,r) . c100a)} is functional non empty complex-functions-membered set
x is complex real ext-real Element of the carrier of R^1
CircleMap " {((2,r) . c100a)} is V195() V196() V197() Element of bool the carrier of K708()
y is non empty complex real integer ext-real non even set
y / 2 is non empty complex real ext-real Element of COMPLEX
y * (2 ") is non empty complex real ext-real set
(((2,r) . c100a),((2,r) . c100a),x,p) is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
hx is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
hx . 1 is complex real ext-real set
hx . 0 is complex real ext-real set
(hx . 0) + y is complex real ext-real set
CircleMap . y is Relation-like Function-like complex-yielding ext-real-valued real-valued set
CircleMap . 0 is Relation-like Function-like complex-yielding ext-real-valued real-valued set
( the carrier of I[01],hx,x) is Relation-like the carrier of I[01] -defined the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total total quasi_total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
- x is complex real ext-real set
(- x) + hx is Relation-like the carrier of I[01] -defined Function-like total complex-yielding ext-real-valued real-valued set
R^1 y is complex real ext-real Element of the carrier of K708()
( the carrier of I[01],hx,x) . 0 is complex real ext-real set
( the carrier of I[01],hx,x) . 1 is complex real ext-real set
j0 is complex real ext-real Element of the carrier of I[01]
hx . j0 is complex real ext-real Element of the carrier of R^1
((hx . j0),x) is complex real ext-real Element of the carrier of R^1
(hx . j0) + (- x) is complex real ext-real set
(x,x) is complex real ext-real Element of the carrier of R^1
x + (- x) is complex real ext-real set
j1 is complex real ext-real Element of the carrier of I[01]
hx . j1 is complex real ext-real Element of the carrier of R^1
((hx . j1),x) is complex real ext-real Element of the carrier of R^1
(hx . j1) + (- x) is complex real ext-real set
(x,y) is complex real ext-real Element of the carrier of R^1
((x,y),x) is complex real ext-real Element of the carrier of R^1
(x,y) + (- x) is complex real ext-real set
c₃₂ is Relation-like the carrier of I[01] -defined the carrier of K708() -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued Path of R^1 0, R^1 y
CircleMap * c₃₂ is Relation-like the carrier of I[01] -defined the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued the carrier of (Tunit_circle 2) -valued Function-like non empty total total quasi_total quasi_total quasi_total continuous Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of CircleMap . (R^1 0),CircleMap . (R^1 y)
CircleMap . (R^1 0) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
CircleMap . (R^1 y) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
I[01] --> x is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
the carrier of I[01] --> x is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like constant non empty total quasi_total quasi_total continuous complex-yielding ext-real-valued real-valued Element of bool [: the carrier of I[01], the carrier of R^1:]
{x} is non empty V195() V196() V197() set
[: the carrier of I[01],{x}:] is Relation-like complex-yielding ext-real-valued real-valued set
X is Relation-like the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued Path of x,x
(x,x,X,x) is Relation-like the carrier of I[01] -defined the carrier of I[01] -defined the carrier of R^1 -valued Function-like non empty total total quasi_total quasi_total continuous complex-yielding ext-real-valued real-valued Path of (x,x),(x,x)
(x,x) is complex real ext-real Element of the carrier of R^1
x + (- x) is complex real ext-real set
(- x) + X is Relation-like the carrier of I[01] -defined Function-like total complex-yielding ext-real-valued real-valued set
CircleMap * (x,x,X,x) is Relation-like the carrier of I[01] -defined the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued the carrier of (Tunit_circle 2) -valued Function-like non empty total total quasi_total quasi_total quasi_total continuous Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of CircleMap . (x,x),CircleMap . (x,x)
CircleMap . (x,x) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
dom Ciso is non empty Element of bool the carrier of INT.Group
bool the carrier of INT.Group is V74() set
Ciso . y is set
Class ((EqRel ((Tunit_circle 2),c[10])),(CircleMap * c₃₂)) is Element of bool K681((Tunit_circle 2),c[10])
bool K681((Tunit_circle 2),c[10]) is set
Ciso . 0 is set
Class ((EqRel ((Tunit_circle 2),c[10])),(CircleMap * (x,x,X,x))) is Element of bool K681((Tunit_circle 2),c[10])
@' 0 is Element of the carrier of INT.Group
Ciso . (@' 0) is Element of the carrier of K684((Tunit_circle 2),c[10])
@' y is Element of the carrier of INT.Group
Ciso . (@' y) is Element of the carrier of K684((Tunit_circle 2),c[10])
Cx is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of c[10] , c[10]
Ch is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Path of c[10] , c[10]
[: the carrier of [:I[01],I[01]:], the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of [:I[01],I[01]:], the carrier of (Tunit_circle 2):] is set
F is Relation-like the carrier of [:I[01],I[01]:] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of [:I[01],I[01]:], the carrier of (Tunit_circle 2):]
S is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
Arg S is complex real ext-real Element of REAL
euc2cpx S is complex Element of COMPLEX
S `1 is complex real ext-real Element of REAL
K382(S,1) is complex real ext-real Element of REAL
S `2 is complex real ext-real Element of REAL
K382(S,2) is complex real ext-real Element of REAL
(S `2) * <i> is complex set
(S `1) + ((S `2) * <i>) is complex set
Arg (euc2cpx S) is complex real ext-real Element of REAL
- (Arg S) is complex real ext-real Element of REAL
(1,(- (Arg S))) is Relation-like the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) -defined the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) -valued Function-like one-to-one non empty total quasi_total onto bijective continuous Function-yielding V178() being_homeomorphism complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (Tcircle ((0. (TOP-REAL 2)),1)):]
the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (Tcircle ((0. (TOP-REAL 2)),1)):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (Tcircle ((0. (TOP-REAL 2)),1)):] is set
Rotate (- (Arg S)) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
(Rotate (- (Arg S))) | (Tcircle ((0. (TOP-REAL 2)),1)) is Relation-like the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (TOP-REAL 2):]
[: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (TOP-REAL 2):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL 2)),1)), the carrier of (TOP-REAL 2):] is set
(Rotate (- (Arg S))) | the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) is Relation-like the carrier of (TOP-REAL 2) -defined the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) -defined the carrier of (TOP-REAL 2) -defined the carrier of (TOP-REAL 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
(1,(- (Arg S))) * F is Relation-like the carrier of [:I[01],I[01]:] -defined the carrier of (Tcircle ((0. (TOP-REAL 2)),1)) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of [:I[01],I[01]:], the carrier of (Tcircle ((0. (TOP-REAL 2)),1)):]
[: the carrier of [:I[01],I[01]:], the carrier of (Tcircle ((0. (TOP-REAL 2)),1)):] is Relation-like set
bool [: the carrier of [:I[01],I[01]:], the carrier of (Tcircle ((0. (TOP-REAL 2)),1)):] is set
|.(euc2cpx S).| is complex real ext-real Element of REAL
Re (euc2cpx S) is complex real ext-real Element of REAL
(Re (euc2cpx S)) ^2 is complex real ext-real non negative Element of REAL
(Re (euc2cpx S)) * (Re (euc2cpx S)) is complex real ext-real non negative set
Im (euc2cpx S) is complex real ext-real Element of REAL
(Im (euc2cpx S)) ^2 is complex real ext-real non negative Element of REAL
(Im (euc2cpx S)) * (Im (euc2cpx S)) is complex real ext-real non negative set
((Re (euc2cpx S)) ^2) + ((Im (euc2cpx S)) ^2) is complex real ext-real non negative Element of REAL
sqrt (((Re (euc2cpx S)) ^2) + ((Im (euc2cpx S)) ^2)) is complex real ext-real non negative Element of REAL
|.S.| is complex real ext-real non negative Element of REAL
sqr S is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued FinSequence of REAL
mlt (S,S) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
K145() .: (S,S) is Relation-like Function-like set
K254(S,S) is Relation-like Function-like set
K254(S,S) (#) K145() is Relation-like REAL -valued Function-like complex-yielding ext-real-valued real-valued set
S (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued set
Sum (sqr S) is complex real ext-real Element of REAL
sqrt (Sum (sqr S)) is complex real ext-real Element of REAL
Rotate ((euc2cpx S),(- (Arg S))) is complex Element of COMPLEX
(- (Arg S)) + (Arg (euc2cpx S)) is complex real ext-real Element of REAL
cos ((- (Arg S)) + (Arg (euc2cpx S))) is complex real ext-real Element of REAL
|.(euc2cpx S).| * (cos ((- (Arg S)) + (Arg (euc2cpx S)))) is complex real ext-real Element of REAL
sin ((- (Arg S)) + (Arg (euc2cpx S))) is complex real ext-real Element of REAL
|.(euc2cpx S).| * (sin ((- (Arg S)) + (Arg (euc2cpx S)))) is complex real ext-real Element of REAL
(|.(euc2cpx S).| * (sin ((- (Arg S)) + (Arg (euc2cpx S))))) * <i> is complex set
(|.(euc2cpx S).| * (cos ((- (Arg S)) + (Arg (euc2cpx S))))) + ((|.(euc2cpx S).| * (sin ((- (Arg S)) + (Arg (euc2cpx S))))) * <i>) is complex set
(1,(- (Arg S))) . ((2,r) . c100a) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(Rotate (- (Arg S))) . S is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
[: the carrier of I[01], the carrier of (Tunit_circle (2 + 1)):] is Relation-like set
bool [: the carrier of I[01], the carrier of (Tunit_circle (2 + 1)):] is set
[: the carrier of I[01], the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of I[01], the carrier of (Tunit_circle 2):] is set
BU is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
t is complex real ext-real Element of the carrier of I[01]
BU . t is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
CircleMap * hx is Relation-like the carrier of I[01] -defined the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued the carrier of (Tunit_circle 2) -valued Function-like non empty total total quasi_total quasi_total continuous Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
H₁(r) . t is Relation-like Function-like complex-yielding ext-real-valued real-valued set
hx . t is complex real ext-real Element of the carrier of R^1
CircleMap . (hx . t) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
((1,(- (Arg S))) * F) . (t,0) is set
j0 is complex real ext-real Element of the carrier of I[01]
F . (t,j0) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(1,(- (Arg S))) . (F . (t,j0)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
T is Relation-like NAT -defined Function-like V32() V39(2) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 2)
(1,(- (Arg S))) . T is Relation-like Function-like complex-yielding ext-real-valued real-valued set
((hx . t),x) is complex real ext-real Element of the carrier of R^1
(hx . t) + (- x) is complex real ext-real set
CircleMap . ((hx . t),x) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
c₃₂ . t is complex real ext-real Element of the carrier of K708()
CircleMap . (c₃₂ . t) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
Ch . t is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
((1,(- (Arg S))) * F) . (t,1) is set
j1 is complex real ext-real Element of the carrier of I[01]
F . (t,j1) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(1,(- (Arg S))) . (F . (t,j1)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
f1 . t is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(1,(- (Arg S))) . (f1 . t) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
CircleMap . (x,x) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
X . t is complex real ext-real Element of the carrier of R^1
((X . t),x) is complex real ext-real Element of the carrier of R^1
(X . t) + (- x) is complex real ext-real set
CircleMap . ((X . t),x) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(x,x,X,x) . t is complex real ext-real Element of the carrier of R^1
CircleMap . ((x,x,X,x) . t) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
Cx . t is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
((1,(- (Arg S))) * F) . (0,t) is set
F . (j0,t) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(1,(- (Arg S))) . (F . (j0,t)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
((1,(- (Arg S))) * F) . (1,t) is set
F . (j1,t) is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
(1,(- (Arg S))) . (F . (j1,t)) is Relation-like Function-like complex-yielding ext-real-valued real-valued set
r is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total continuous Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (2 + 1))),1)), the carrier of (TOP-REAL 2):]
(2,r) is Relation-like the carrier of (Tunit_circle (2 + 1)) -defined the carrier of (Tunit_circle 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle (2 + 1)), the carrier of (Tunit_circle 2):]
(2,r) * (1) is Relation-like the carrier of I[01] -defined the carrier of (Tunit_circle 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of I[01], the carrier of (Tunit_circle 2):]
(2,r) . c100a is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Tunit_circle 2)
r is non empty complex ordinal natural real integer ext-real positive non negative set
r + 1 is non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() Element of NAT
TOP-REAL (r + 1) is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL (r + 1)) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
f is Relation-like NAT -defined Function-like V32() V39(r + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
p is non empty complex real ext-real non positive negative set
Tcircle (f,p) is TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
Sphere (f,p) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V32() open closed boundary nowhere_dense FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178() V195() V196() V197() V198() V199() V200() V201() V214( TOP-REAL (r + 1)) complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued Element of bool the carrier of (TOP-REAL (r + 1))
bool the carrier of (TOP-REAL (r + 1)) is V74() set
(TOP-REAL (r + 1)) | (Sphere (f,p)) is empty trivial V47() {} -element strict TopSpace-like T_0 V245( TOP-REAL (r + 1)) complex-functions-membered real-functions-membered SubSpace of TOP-REAL (r + 1)
the carrier of (Tcircle (f,p)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
TOP-REAL r is non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group RLTopStruct
the carrier of (TOP-REAL r) is functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle (f,p)), the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of (Tcircle (f,p)), the carrier of (TOP-REAL r):] is set
e is Relation-like the carrier of (Tcircle (f,p)) -defined the carrier of (TOP-REAL r) -valued Function-like total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle (f,p)), the carrier of (TOP-REAL r):]
f1 is Relation-like NAT -defined Function-like V32() V39(r + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
h is Relation-like NAT -defined Function-like V32() V39(r + 1) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL (r + 1))
dom e is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle (f,p))
bool the carrier of (Tcircle (f,p)) is set
p is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
r is complex real ext-real non negative set
Tcircle (p,r) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 3
Sphere (p,r) is functional non empty closed V214( TOP-REAL 3) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 3)
(TOP-REAL 3) | (Sphere (p,r)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 3
the carrier of (Tcircle (p,r)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle (p,r)), the carrier of (TOP-REAL 2):] is Relation-like set
bool [: the carrier of (Tcircle (p,r)), the carrier of (TOP-REAL 2):] is set
f is Relation-like the carrier of (Tcircle (p,r)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle (p,r)), the carrier of (TOP-REAL 2):]
dom f is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle (p,r))
bool the carrier of (Tcircle (p,r)) is V74() set
e is non empty complex real ext-real positive non negative set
Tcircle (p,e) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 3
Sphere (p,e) is functional non empty closed V214( TOP-REAL 3) complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 3)
(TOP-REAL 3) | (Sphere (p,e)) is non empty strict TopSpace-like complex-functions-membered real-functions-membered SubSpace of TOP-REAL 3
the carrier of (Tcircle (p,e)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle (p,e)), the carrier of (TOP-REAL 2):] is Relation-like set
bool [: the carrier of (Tcircle (p,e)), the carrier of (TOP-REAL 2):] is set
the carrier of (Tcircle ((0. (TOP-REAL 3)),1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL 3)),1)), the carrier of (Tcircle (p,e)):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL 3)),1)), the carrier of (Tcircle (p,e)):] is set
(3,p,e) is Relation-like the carrier of (Tunit_circle 3) -defined the carrier of (Tcircle (p,e)) -valued Function-like one-to-one non empty total quasi_total onto bijective continuous Function-yielding V178() being_homeomorphism complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tunit_circle 3), the carrier of (Tcircle (p,e)):]
[: the carrier of (Tunit_circle 3), the carrier of (Tcircle (p,e)):] is Relation-like set
bool [: the carrier of (Tunit_circle 3), the carrier of (Tcircle (p,e)):] is set
h is Relation-like the carrier of (Tcircle ((0. (TOP-REAL 3)),1)) -defined the carrier of (Tcircle (p,e)) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL 3)),1)), the carrier of (Tcircle (p,e)):]
f1 is Relation-like the carrier of (Tcircle (p,e)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total continuous Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle (p,e)), the carrier of (TOP-REAL 2):]
f1 * h is Relation-like the carrier of (Tcircle ((0. (TOP-REAL 3)),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL 3)),1)), the carrier of (TOP-REAL 2):]
[: the carrier of (Tcircle ((0. (TOP-REAL 3)),1)), the carrier of (TOP-REAL 2):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL 3)),1)), the carrier of (TOP-REAL 2):] is set
x is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
y is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
f * h is Relation-like the carrier of (Tcircle ((0. (TOP-REAL 3)),1)) -defined the carrier of (TOP-REAL 2) -valued Function-like Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL 3)),1)), the carrier of (TOP-REAL 2):]
dom (f * h) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL 3)),1))
bool the carrier of (Tcircle ((0. (TOP-REAL 3)),1)) is V74() set
(f * h) . x is Relation-like Function-like complex-yielding ext-real-valued real-valued set
(f * h) . y is Relation-like Function-like complex-yielding ext-real-valued real-valued set
h . x is Relation-like Function-like complex-yielding ext-real-valued real-valued set
h . y is Relation-like Function-like complex-yielding ext-real-valued real-valued set
hx is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
hy is Relation-like NAT -defined Function-like V32() V39(3) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued Element of the carrier of (TOP-REAL 3)
dom f is non empty set
f . hx is Relation-like Function-like complex-yielding ext-real-valued real-valued set
f . hy is Relation-like Function-like complex-yielding ext-real-valued real-valued set
e is Relation-like Function-like complex-yielding ext-real-valued real-valued Element of the carrier of (Euclid 3)
Sphere (e,r) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Euclid 3)
bool the carrier of (Euclid 3) is V74() set
{e} is functional non empty complex-functions-membered set
LSeg (p,p) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL 3)
{ (((1 - b₁) * p) + (b₁ * p)) where b₁ is complex real ext-real Element of REAL : ( 0 <= b₁ & b₁ <= 1 ) } is set
dom f is non empty set
f . p is Relation-like Function-like complex-yielding ext-real-valued real-valued set