:: TOPREALB semantic presentation

REAL is non empty V27() V67() V68() V69() V73() non with_non-empty_elements set
NAT is V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements Element of K6(REAL)
K6(REAL) is non empty set
K744() is non empty strict TopSpace-like V299() SubSpace of R^1
R^1 is non empty strict TopSpace-like V299() TopStruct
the carrier of K744() is non empty V67() V68() V69() set
COMPLEX is non empty V27() V67() V73() set
omega is V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements set
K6(omega) is non empty set
K7(REAL,REAL) is non empty V51() V52() V53() set
K6(K7(REAL,REAL)) is non empty set
K6(NAT) is non empty set
RAT is non empty V27() V67() V68() V69() V70() V73() set
INT is non empty V27() V67() V68() V69() V70() V71() V73() set
K7(NAT,REAL) is V51() V52() V53() set
K6(K7(NAT,REAL)) is non empty set
K7(NAT,COMPLEX) is V51() set
K6(K7(NAT,COMPLEX)) is non empty set
K7(COMPLEX,COMPLEX) is non empty V51() set
K6(K7(COMPLEX,COMPLEX)) is non empty set
{} is empty trivial Function-like functional V67() V68() V69() V70() V71() V72() V73() set
the empty trivial Function-like functional V67() V68() V69() V70() V71() V72() V73() set is empty trivial Function-like functional V67() V68() V69() V70() V71() V72() V73() set
1 is non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
{{},1} is non empty set
K562() is non empty V138() L9()
the carrier of K562() is non empty set
K567() is non empty V138() V201() V202() V203() V205() V231() V232() V233() V234() V235() V236() L9()
K568() is non empty V138() V203() V205() V234() V235() V236() M31(K567())
K569() is non empty V138() V201() V203() V205() V234() V235() V236() V237() M34(K567(),K568())
K571() is non empty V138() V201() V203() V205() L9()
K572() is non empty V138() V201() V203() V205() V237() M31(K571())
K7(1,1) is non empty V20( RAT ) V20( INT ) V51() V52() V53() V54() set
K6(K7(1,1)) is non empty set
K7(K7(1,1),1) is non empty V20( RAT ) V20( INT ) V51() V52() V53() V54() set
K6(K7(K7(1,1),1)) is non empty set
K7(K7(1,1),REAL) is non empty V51() V52() V53() set
K6(K7(K7(1,1),REAL)) is non empty set
K7(K7(REAL,REAL),REAL) is non empty V51() V52() V53() set
K6(K7(K7(REAL,REAL),REAL)) is non empty set
2 is non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
K7(2,2) is non empty V20( RAT ) V20( INT ) V51() V52() V53() V54() set
K7(K7(2,2),REAL) is non empty V51() V52() V53() set
K6(K7(K7(2,2),REAL)) is non empty set
TOP-REAL 2 is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
the carrier of (TOP-REAL 2) is non empty functional set
K6( the carrier of (TOP-REAL 2)) is non empty set
K7( the carrier of (TOP-REAL 2),REAL) is non empty V51() V52() V53() set
K6(K7( the carrier of (TOP-REAL 2),REAL)) is non empty set
K737() is non empty strict TopSpace-like V299() TopStruct
the carrier of K737() is non empty V67() V68() V69() set
RealSpace is strict V299() MetrStruct
[:K744(),K744():] is non empty strict TopSpace-like TopStruct
the carrier of [:K744(),K744():] is non empty set
[:R^1,R^1:] is non empty strict TopSpace-like TopStruct
the carrier of [:R^1,R^1:] is non empty set
K7( the carrier of [:R^1,R^1:], the carrier of (TOP-REAL 2)) is non empty set
K6(K7( the carrier of [:R^1,R^1:], the carrier of (TOP-REAL 2))) is non empty set
K7(K7(COMPLEX,COMPLEX),COMPLEX) is non empty V51() set
K6(K7(K7(COMPLEX,COMPLEX),COMPLEX)) is non empty set
K7(RAT,RAT) is non empty V20( RAT ) V51() V52() V53() set
K6(K7(RAT,RAT)) is non empty set
K7(K7(RAT,RAT),RAT) is non empty V20( RAT ) V51() V52() V53() set
K6(K7(K7(RAT,RAT),RAT)) is non empty set
K7(INT,INT) is non empty V20( RAT ) V20( INT ) V51() V52() V53() set
K6(K7(INT,INT)) is non empty set
K7(K7(INT,INT),INT) is non empty V20( RAT ) V20( INT ) V51() V52() V53() set
K6(K7(K7(INT,INT),INT)) is non empty set
K7(NAT,NAT) is V20( RAT ) V20( INT ) V51() V52() V53() V54() set
K7(K7(NAT,NAT),NAT) is V20( RAT ) V20( INT ) V51() V52() V53() V54() set
K6(K7(K7(NAT,NAT),NAT)) is non empty set
0 is empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() Element of NAT
sin is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
cos is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom sin is V67() V68() V69() Element of K6(REAL)
dom cos is V67() V68() V69() Element of K6(REAL)
cos 0 is complex real ext-real Element of REAL
sin 0 is complex real ext-real Element of REAL
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
cos (PI / 2) is complex real ext-real Element of REAL
sin (PI / 2) is complex real ext-real Element of REAL
cos PI is complex real ext-real Element of REAL
- 1 is non empty complex real ext-real non positive negative integer Element of REAL
sin PI is complex real ext-real Element of REAL
PI + (PI / 2) is non empty complex real ext-real positive non negative Element of REAL
cos (PI + (PI / 2)) is complex real ext-real Element of REAL
sin (PI + (PI / 2)) is complex real ext-real Element of REAL
2 * PI is non empty complex real ext-real positive non negative Element of REAL
cos (2 * PI) is complex real ext-real Element of REAL
sin (2 * PI) is complex real ext-real Element of REAL
0. (TOP-REAL 2) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like zero Element of the carrier of (TOP-REAL 2)
the ZeroF of (TOP-REAL 2) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
sqrt 1 is complex real ext-real Element of REAL
sqrt 2 is complex real ext-real Element of REAL
- 1 is non empty complex real ext-real non positive negative integer set
Euclid 2 is non empty strict Reflexive discerning V245() triangle MetrStruct
the carrier of (Euclid 2) is non empty set
the carrier of R^1 is non empty V67() V68() V69() set
I(01) is strict TopSpace-like V299() SubSpace of K744()
K6( the carrier of R^1) is non empty set
K7( the carrier of R^1, the carrier of R^1) is non empty V51() V52() V53() set
K6(K7( the carrier of R^1, the carrier of R^1)) is non empty set
[.0,PI.] is non empty closed V67() V68() V69() Element of K6(REAL)
].PI,(2 * PI).[ is open V67() V68() V69() Element of K6(REAL)
- (PI / 2) is non empty complex real ext-real non positive negative Element of REAL
[.(- (PI / 2)),(PI / 2).] is non empty closed V67() V68() V69() Element of K6(REAL)
3 is non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
3 / 2 is non empty complex real ext-real positive non negative Element of REAL
(3 / 2) * PI is non empty complex real ext-real positive non negative Element of REAL
].(PI / 2),((3 / 2) * PI).[ is open V67() V68() V69() Element of K6(REAL)
[.(PI / 2),((3 / 2) * PI).] is closed V67() V68() V69() Element of K6(REAL)
].((3 / 2) * PI),(2 * PI).[ is open V67() V68() V69() Element of K6(REAL)
rng sin is V67() V68() V69() set
[.(- 1),1.] is non empty closed V67() V68() V69() Element of K6(REAL)
rng cos is V67() V68() V69() set
R^2-unit_square is non empty functional compact being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
(TOP-REAL 2) | R^2-unit_square is non empty strict TopSpace-like compact V221() V222() V266() SubSpace of TOP-REAL 2
the carrier of ((TOP-REAL 2) | R^2-unit_square) is non empty set
arcsin is V16() V19( REAL ) V20( REAL ) Function-like one-to-one V51() V52() V53() Element of K6(K7(REAL,REAL))
dom arcsin is V67() V68() V69() Element of K6(REAL)
arcsin | [.(- 1),1.] is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
arccos is V16() V19( REAL ) V20( REAL ) Function-like one-to-one V51() V52() V53() Element of K6(K7(REAL,REAL))
rng arccos is V67() V68() V69() Element of K6(REAL)
dom arccos is V67() V68() V69() Element of K6(REAL)
arccos (- 1) is complex real ext-real Element of REAL
arccos 1 is complex real ext-real Element of REAL
arccos | [.(- 1),1.] is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
proj1 is non empty V16() V19( the carrier of (TOP-REAL 2)) V20( REAL ) Function-like total quasi_total V51() V52() V53() continuous Element of K6(K7( the carrier of (TOP-REAL 2),REAL))
R^1 is non empty strict TopSpace-like V299() V303() SubSpace of R^1
the carrier of R^1 is non empty V67() V68() V69() set
Closed-Interval-TSpace ((- 1),1) is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (Closed-Interval-TSpace ((- 1),1)) is non empty V67() V68() V69() set
1 - 0 is non empty complex real ext-real positive non negative integer Element of REAL
1 / 2 is non empty complex real ext-real positive non negative Element of REAL
(3 / 2) - (1 / 2) is complex real ext-real Element of REAL
].0,1.[ is non empty open V67() V68() V69() Element of K6(REAL)
].(1 / 2),(3 / 2).[ is open V67() V68() V69() Element of K6(REAL)
p1 is non empty complex real ext-real positive non negative set
(1 / 2) + p1 is non empty complex real ext-real positive non negative Element of REAL
].(1 / 2),((1 / 2) + p1).[ is non empty open V67() V68() V69() Element of K6(REAL)
PI / 1 is non empty complex real ext-real positive non negative Element of REAL
1 * PI is non empty complex real ext-real positive non negative Element of REAL
A is non empty TopSpace-like TopStruct
the carrier of A is non empty set
TUC is non empty TopSpace-like SubSpace of A
cS1 is non empty TopSpace-like SubSpace of TUC
the carrier of cS1 is non empty set
TREC is Element of the carrier of cS1
the carrier of TUC is non empty set
A is TopSpace-like TopStruct
the carrier of A is set
K6( the carrier of A) is non empty set
TUC is TopSpace-like SubSpace of A
cS1 is TopSpace-like SubSpace of TUC
the carrier of cS1 is set
K6( the carrier of cS1) is non empty set
TREC is Element of K6( the carrier of cS1)
the carrier of TUC is set
K6( the carrier of TUC) is non empty set
A is complex real ext-real set
TUC is complex real ext-real set
A * TUC is complex real ext-real set
cS1 is complex real ext-real set
(A * TUC) + cS1 is complex real ext-real set
sin ((A * TUC) + cS1) is complex real ext-real set
AffineMap (A,cS1) is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
sin * (AffineMap (A,cS1)) is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
(sin * (AffineMap (A,cS1))) . TUC is complex real ext-real set
sin . ((A * TUC) + cS1) is complex real ext-real set
(AffineMap (A,cS1)) . TUC is complex real ext-real set
sin . ((AffineMap (A,cS1)) . TUC) is complex real ext-real set
A is complex real ext-real set
TUC is complex real ext-real set
A * TUC is complex real ext-real set
cS1 is complex real ext-real set
(A * TUC) + cS1 is complex real ext-real set
cos ((A * TUC) + cS1) is complex real ext-real set
AffineMap (A,cS1) is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
cos * (AffineMap (A,cS1)) is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
(cos * (AffineMap (A,cS1))) . TUC is complex real ext-real set
cos . ((A * TUC) + cS1) is complex real ext-real set
(AffineMap (A,cS1)) . TUC is complex real ext-real set
cos . ((AffineMap (A,cS1)) . TUC) is complex real ext-real set
A is non empty complex real ext-real set
TUC is complex real ext-real set
AffineMap (A,TUC) is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
rng (AffineMap (A,TUC)) is V67() V68() V69() Element of K6(REAL)
A is complex real ext-real set
TUC is complex real ext-real set
{ ].(A + b₁),(TUC + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
cS1 is set
A is complex real ext-real set
TUC is complex real ext-real set
(A,TUC) is set
{ ].(A + b₁),(TUC + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
TOUCm is set
TREC is complex real ext-real set
X is complex real ext-real integer V66() Element of INT
TREC + X is complex real ext-real set
TOUC is complex real ext-real set
TOUC + X is complex real ext-real set
].(TREC + X),(TOUC + X).[ is open V67() V68() V69() Element of K6(REAL)
(TREC,TOUC) is set
{ ].(TREC + b₁),(TOUC + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
A is complex real ext-real set
TUC is complex real ext-real set
(A,TUC) is set
{ ].(A + b₁),(TUC + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
A + 0 is complex real ext-real Element of REAL
TUC + 0 is complex real ext-real Element of REAL
].(A + 0),(TUC + 0).[ is open V67() V68() V69() Element of K6(REAL)
A is complex real ext-real set
TUC is complex real ext-real set
A - TUC is complex real ext-real set
(TUC,A) is non empty set
{ ].(TUC + b₁),(A + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
TUC + 1 is complex real ext-real Element of REAL
(TUC + 1) + 0 is complex real ext-real Element of REAL
cS1 is ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
(TUC + 1) + cS1 is complex real ext-real Element of REAL
(A - TUC) + TUC is complex real ext-real set
1 + TUC is complex real ext-real Element of REAL
cS1 + 1 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
TUC + (cS1 + 1) is complex real ext-real Element of REAL
cS1 is set
TREC is set
TOUC is complex real ext-real integer V66() Element of INT
TUC + TOUC is complex real ext-real set
A + TOUC is complex real ext-real set
].(TUC + TOUC),(A + TOUC).[ is open V67() V68() V69() Element of K6(REAL)
TOUCm is complex real ext-real integer V66() Element of INT
TUC + TOUCm is complex real ext-real set
A + TOUCm is complex real ext-real set
].(TUC + TOUCm),(A + TOUCm).[ is open V67() V68() V69() Element of K6(REAL)
X is set
Xm is complex real ext-real Element of REAL
TOUC + 1 is complex real ext-real integer Element of REAL
TOUCm - (TOUC + 1) is complex real ext-real integer Element of REAL
(TUC + TOUC) + 1 is complex real ext-real Element of REAL
Y is ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
((TUC + TOUC) + 1) + Y is complex real ext-real Element of REAL
(A + TOUC) - TOUC is complex real ext-real set
Y + 1 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
(TUC + TOUC) + (Y + 1) is complex real ext-real Element of REAL
((TUC + TOUC) + (Y + 1)) - TOUC is complex real ext-real Element of REAL
TUC + (Y + 1) is complex real ext-real Element of REAL
TOUCm + 1 is complex real ext-real integer Element of REAL
TOUC - (TOUCm + 1) is complex real ext-real integer Element of REAL
(TUC + TOUCm) + 1 is complex real ext-real Element of REAL
Y is ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
((TUC + TOUCm) + 1) + Y is complex real ext-real Element of REAL
(A + TOUCm) - TOUCm is complex real ext-real set
Y + 1 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
(TUC + TOUCm) + (Y + 1) is complex real ext-real Element of REAL
((TUC + TOUCm) + (Y + 1)) - TOUCm is complex real ext-real Element of REAL
TUC + (Y + 1) is complex real ext-real Element of REAL
A is complex real ext-real set
TUC is complex real ext-real set
(A,TUC) is non empty set
{ ].(A + b₁),(TUC + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
K6( the carrier of R^1) is non empty set
K6(K6( the carrier of R^1)) is non empty set
bool the carrier of R^1 is non empty Element of K6(K6( the carrier of R^1))
cS1 is set
TREC is complex real ext-real integer V66() Element of INT
A + TREC is complex real ext-real set
TUC + TREC is complex real ext-real set
].(A + TREC),(TUC + TREC).[ is open V67() V68() V69() Element of K6(REAL)
A is complex real ext-real set
TUC is complex real ext-real set
(A,TUC) is non empty set
{ ].(A + b₁),(TUC + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
(A,TUC) is non empty Element of K6(K6( the carrier of R^1))
cS1 is V67() V68() V69() Element of K6( the carrier of R^1)
TREC is complex real ext-real integer V66() Element of INT
A + TREC is complex real ext-real set
TUC + TREC is complex real ext-real set
].(A + TREC),(TUC + TREC).[ is open V67() V68() V69() Element of K6(REAL)
A is complex real ext-real set
A is V67() V68() V69() Element of K6(REAL)
A is non empty V67() V68() V69() Element of K6(REAL)
(A) is V67() V68() V69() Element of K6( the carrier of R^1)
A is open V67() V68() V69() Element of K6(REAL)
(A) is V67() V68() V69() Element of K6( the carrier of R^1)
A is closed V67() V68() V69() Element of K6(REAL)
(A) is V67() V68() V69() Element of K6( the carrier of R^1)
A is open V67() V68() V69() Element of K6(REAL)
(A) is V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (A) is strict TopSpace-like V299() SubSpace of R^1
TUC is V67() V68() V69() Element of K6( the carrier of R^1)
the carrier of (R^1 | (A)) is V67() V68() V69() set
A is closed V67() V68() V69() Element of K6(REAL)
(A) is V67() V68() V69() closed Element of K6( the carrier of R^1)
R^1 | (A) is strict TopSpace-like V299() SubSpace of R^1
TUC is V67() V68() V69() Element of K6( the carrier of R^1)
the carrier of (R^1 | (A)) is V67() V68() V69() set
A is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
dom A is V67() V68() V69() Element of K6(REAL)
((dom A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom A)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom A))) is V67() V68() V69() set
rng A is V67() V68() V69() Element of K6(REAL)
((rng A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng A)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng A))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A)))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A))))) is non empty set
A is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
rng A is V67() V68() V69() Element of K6(REAL)
((rng A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng A)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng A))) is V67() V68() V69() set
(A) is V16() V19( the carrier of (R^1 | ((dom A)))) V20( the carrier of (R^1 | ((rng A)))) Function-like quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A)))))
dom A is V67() V68() V69() Element of K6(REAL)
((dom A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom A)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom A))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A)))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A))))) is non empty set
rng (A) is V67() V68() V69() Element of K6( the carrier of (R^1 | ((rng A))))
K6( the carrier of (R^1 | ((rng A)))) is non empty set
A is V16() V19( REAL ) V20( REAL ) Function-like one-to-one V51() V52() V53() Element of K6(K7(REAL,REAL))
(A) is V16() V19( the carrier of (R^1 | ((dom A)))) V20( the carrier of (R^1 | ((rng A)))) Function-like quasi_total onto V51() V52() V53() Element of K6(K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A)))))
dom A is V67() V68() V69() Element of K6(REAL)
((dom A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom A)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom A))) is V67() V68() V69() set
rng A is V67() V68() V69() Element of K6(REAL)
((rng A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng A)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng A))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A)))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A))))) is non empty set
[#] REAL is non empty non proper V67() V68() V69() Element of K6(REAL)
(([#] REAL)) is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | (([#] REAL)) is non empty strict TopSpace-like V299() SubSpace of R^1
[#] R^1 is non empty non proper V67() V68() V69() open closed dense non boundary Element of K6( the carrier of R^1)
A is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
dom A is V67() V68() V69() Element of K6(REAL)
((dom A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom A)) is strict TopSpace-like V299() SubSpace of R^1
[#] R^1 is non empty non proper V67() V68() V69() open closed dense non boundary Element of K6( the carrier of R^1)
A is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total V51() V52() V53() Element of K6(K7(REAL,REAL))
rng A is V67() V68() V69() Element of K6(REAL)
((rng A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng A)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng A))) is V67() V68() V69() set
K7( the carrier of R^1, the carrier of (R^1 | ((rng A)))) is V51() V52() V53() set
K6(K7( the carrier of R^1, the carrier of (R^1 | ((rng A))))) is non empty set
dom A is V67() V68() V69() Element of K6(REAL)
((dom A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom A)) is strict TopSpace-like V299() SubSpace of R^1
(A) is V16() V19( the carrier of (R^1 | ((dom A)))) V20( the carrier of (R^1 | ((rng A)))) Function-like quasi_total onto V51() V52() V53() Element of K6(K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A)))))
the carrier of (R^1 | ((dom A))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A)))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A))))) is non empty set
K7( the carrier of R^1, the carrier of R^1) is non empty V51() V52() V53() set
K6(K7( the carrier of R^1, the carrier of R^1)) is non empty set
A is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total V51() V52() V53() Element of K6(K7(REAL,REAL))
dom A is V67() V68() V69() Element of K6(REAL)
rng A is V67() V68() V69() Element of K6(REAL)
TUC is non empty V67() V68() V69() Element of K6(REAL)
(TUC) is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | (TUC) is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | (TUC)) is non empty V67() V68() V69() set
K7( the carrier of R^1, the carrier of (R^1 | (TUC))) is non empty V51() V52() V53() set
K6(K7( the carrier of R^1, the carrier of (R^1 | (TUC)))) is non empty set
(sin) is V16() V19( the carrier of (R^1 | ((dom sin)))) V20( the carrier of (R^1 | ((rng sin)))) Function-like quasi_total onto V51() V52() V53() Element of K6(K7( the carrier of (R^1 | ((dom sin))), the carrier of (R^1 | ((rng sin)))))
((dom sin)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom sin)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom sin))) is V67() V68() V69() set
rng sin is V67() V68() V69() Element of K6(REAL)
((rng sin)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng sin)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng sin))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom sin))), the carrier of (R^1 | ((rng sin)))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom sin))), the carrier of (R^1 | ((rng sin))))) is non empty set
(cos) is V16() V19( the carrier of (R^1 | ((dom cos)))) V20( the carrier of (R^1 | ((rng cos)))) Function-like quasi_total onto V51() V52() V53() Element of K6(K7( the carrier of (R^1 | ((dom cos))), the carrier of (R^1 | ((rng cos)))))
((dom cos)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom cos)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom cos))) is V67() V68() V69() set
rng cos is V67() V68() V69() Element of K6(REAL)
((rng cos)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng cos)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng cos))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom cos))), the carrier of (R^1 | ((rng cos)))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom cos))), the carrier of (R^1 | ((rng cos))))) is non empty set
A is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom A is V67() V68() V69() Element of K6(REAL)
((dom A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom A)) is strict TopSpace-like V299() SubSpace of R^1
rng A is V67() V68() V69() Element of K6(REAL)
((rng A)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng A)) is strict TopSpace-like V299() SubSpace of R^1
(A) is V16() V19( the carrier of (R^1 | ((dom A)))) V20( the carrier of (R^1 | ((rng A)))) Function-like quasi_total onto V51() V52() V53() Element of K6(K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A)))))
the carrier of (R^1 | ((dom A))) is V67() V68() V69() set
the carrier of (R^1 | ((rng A))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A)))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom A))), the carrier of (R^1 | ((rng A))))) is non empty set
cS1 is non empty V67() V68() V69() Element of K6(REAL)
(cS1) is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | (cS1) is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | (cS1)) is non empty V67() V68() V69() set
TREC is non empty V67() V68() V69() Element of K6(REAL)
(TREC) is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | (TREC) is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | (TREC)) is non empty V67() V68() V69() set
K7( the carrier of (R^1 | (cS1)), the carrier of (R^1 | (TREC))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | (cS1)), the carrier of (R^1 | (TREC)))) is non empty set
K7( the carrier of (R^1 | (cS1)), the carrier of R^1) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | (cS1)), the carrier of R^1)) is non empty set
TOUC is non empty V16() V19( the carrier of (R^1 | (cS1))) V20( the carrier of (R^1 | (TREC))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | (cS1)), the carrier of (R^1 | (TREC))))
TOUCm is non empty V16() V19( the carrier of (R^1 | (cS1))) V20( the carrier of R^1) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | (cS1)), the carrier of R^1))
X is complex real ext-real Element of the carrier of (R^1 | (cS1))
TOUCm . X is complex real ext-real Element of the carrier of R^1
Xm is non empty V67() V68() V69() a_neighborhood of TOUCm . X
A . X is complex real ext-real set
Y is open V67() V68() V69() Neighbourhood of A . X
A1 is open V67() V68() V69() Neighbourhood of X
A .: A1 is V67() V68() V69() Element of K6(REAL)
C is complex real ext-real set
X - C is complex real ext-real set
X + C is complex real ext-real set
].(X - C),(X + C).[ is open V67() V68() V69() Element of K6(REAL)
X + 0 is complex real ext-real Element of REAL
K6( the carrier of (R^1 | (cS1))) is non empty set
Cm is V67() V68() V69() open Element of K6( the carrier of R^1)
[#] (R^1 | (cS1)) is non empty non proper V67() V68() V69() open closed dense non boundary Element of K6( the carrier of (R^1 | (cS1)))
Cm /\ ([#] (R^1 | (cS1))) is V67() V68() V69() Element of K6( the carrier of (R^1 | (cS1)))
Int Cm is V67() V68() V69() open Element of K6( the carrier of R^1)
X - 0 is complex real ext-real Element of REAL
Ym is complex real ext-real Element of the carrier of R^1
A is V67() V68() V69() Element of K6( the carrier of (R^1 | (cS1)))
Q is non empty V67() V68() V69() open a_neighborhood of X
TOUCm .: Q is V67() V68() V69() Element of K6( the carrier of R^1)
TOUCm .: Cm is V67() V68() V69() Element of K6( the carrier of R^1)
(].0,1.[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
TUC is non empty complex real ext-real set
cS1 is complex real ext-real set
AffineMap (TUC,cS1) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
rng (AffineMap (TUC,cS1)) is V67() V68() V69() Element of K6(REAL)
[#] R^1 is non empty non proper V67() V68() V69() open closed dense non boundary Element of K6( the carrier of R^1)
dom (AffineMap (TUC,cS1)) is V67() V68() V69() Element of K6(REAL)
((dom (AffineMap (TUC,cS1)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom (AffineMap (TUC,cS1)))) is strict TopSpace-like V299() SubSpace of R^1
((rng (AffineMap (TUC,cS1)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng (AffineMap (TUC,cS1)))) is strict TopSpace-like V299() SubSpace of R^1
TUC is non empty complex real ext-real set
cS1 is complex real ext-real set
AffineMap (TUC,cS1) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (AffineMap (TUC,cS1)) is V67() V68() V69() Element of K6(REAL)
((dom (AffineMap (TUC,cS1)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom (AffineMap (TUC,cS1)))) is strict TopSpace-like V299() SubSpace of R^1
rng (AffineMap (TUC,cS1)) is V67() V68() V69() Element of K6(REAL)
((rng (AffineMap (TUC,cS1)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng (AffineMap (TUC,cS1)))) is strict TopSpace-like V299() SubSpace of R^1
((AffineMap (TUC,cS1))) is V16() V19( the carrier of (R^1 | ((dom (AffineMap (TUC,cS1)))))) V20( the carrier of (R^1 | ((rng (AffineMap (TUC,cS1)))))) Function-like one-to-one quasi_total onto bijective V51() V52() V53() continuous Element of K6(K7( the carrier of (R^1 | ((dom (AffineMap (TUC,cS1))))), the carrier of (R^1 | ((rng (AffineMap (TUC,cS1)))))))
the carrier of (R^1 | ((dom (AffineMap (TUC,cS1))))) is V67() V68() V69() set
the carrier of (R^1 | ((rng (AffineMap (TUC,cS1))))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom (AffineMap (TUC,cS1))))), the carrier of (R^1 | ((rng (AffineMap (TUC,cS1)))))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom (AffineMap (TUC,cS1))))), the carrier of (R^1 | ((rng (AffineMap (TUC,cS1))))))) is non empty set
K6( the carrier of (R^1 | ((dom (AffineMap (TUC,cS1)))))) is non empty set
TREC is V67() V68() V69() Element of K6( the carrier of (R^1 | ((dom (AffineMap (TUC,cS1))))))
((AffineMap (TUC,cS1))) .: TREC is V67() V68() V69() Element of K6( the carrier of (R^1 | ((rng (AffineMap (TUC,cS1))))))
K6( the carrier of (R^1 | ((rng (AffineMap (TUC,cS1)))))) is non empty set
TUC is TopSpace-like V221() V222() SubSpace of TOP-REAL 2
the carrier of TUC is set
cS1 is non empty functional compact being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
[#] TUC is non proper open closed dense Element of K6( the carrier of TUC)
K6( the carrier of TUC) is non empty set
cS1 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
TUC is non empty complex real ext-real positive non negative set
Sphere (cS1,TUC) is non empty functional closed V258( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - cS1).| = TUC } is set
cS1 `1 is complex real ext-real Element of REAL
cS1 `2 is complex real ext-real Element of REAL
|[(cS1 `1),(cS1 `2)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
TREC is Element of the carrier of (Euclid 2)
Sphere (TREC,TUC) is Element of K6( the carrier of (Euclid 2))
K6( the carrier of (Euclid 2)) is non empty set
circle ((cS1 `1),(cS1 `2),TUC) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - |[(cS1 `1),(cS1 `2)]|).| = TUC } is set
TUC is ordinal natural complex real ext-real non negative integer set
TOP-REAL TUC is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
the carrier of (TOP-REAL TUC) is non empty functional set
cS1 is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TUC)
TREC is complex real ext-real set
Sphere (cS1,TREC) is functional closed Element of K6( the carrier of (TOP-REAL TUC))
K6( the carrier of (TOP-REAL TUC)) is non empty set
{ b₁ where b₁ is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TUC) : |.(b₁ - cS1).| = TREC } is set
(TOP-REAL TUC) | (Sphere (cS1,TREC)) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL TUC
TUC is non empty ordinal natural complex real ext-real positive non negative integer set
TOP-REAL TUC is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
the carrier of (TOP-REAL TUC) is non empty functional set
cS1 is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TUC)
TREC is complex real ext-real non negative set
(TUC,cS1,TREC) is TopSpace-like V221() V222() SubSpace of TOP-REAL TUC
Sphere (cS1,TREC) is non empty functional closed Element of K6( the carrier of (TOP-REAL TUC))
K6( the carrier of (TOP-REAL TUC)) is non empty set
{ b₁ where b₁ is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TUC) : |.(b₁ - cS1).| = TREC } is set
(TOP-REAL TUC) | (Sphere (cS1,TREC)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TUC
TUC is ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
TOP-REAL TUC is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
the carrier of (TOP-REAL TUC) is non empty functional set
cS1 is complex real ext-real set
TREC is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TUC)
(TUC,TREC,cS1) is TopSpace-like V221() V222() SubSpace of TOP-REAL TUC
Sphere (TREC,cS1) is functional closed V258( TOP-REAL TUC) Element of K6( the carrier of (TOP-REAL TUC))
K6( the carrier of (TOP-REAL TUC)) is non empty set
{ b₁ where b₁ is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TUC) : |.(b₁ - TREC).| = cS1 } is set
(TOP-REAL TUC) | (Sphere (TREC,cS1)) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL TUC
the carrier of (TUC,TREC,cS1) is set
[#] (TUC,TREC,cS1) is non proper open closed dense Element of K6( the carrier of (TUC,TREC,cS1))
K6( the carrier of (TUC,TREC,cS1)) is non empty set
TUC is ordinal natural complex real ext-real non negative integer set
TOP-REAL TUC is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
the carrier of (TOP-REAL TUC) is non empty functional set
cS1 is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TUC)
TREC is empty trivial complex real ext-real non positive non negative Function-like functional V67() V68() V69() V70() V71() V72() V73() set
(TUC,cS1,TREC) is TopSpace-like V221() V222() SubSpace of TOP-REAL TUC
Sphere (cS1,TREC) is functional closed Element of K6( the carrier of (TOP-REAL TUC))
K6( the carrier of (TOP-REAL TUC)) is non empty set
{ b₁ where b₁ is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TUC) : |.(b₁ - cS1).| = TREC } is set
(TOP-REAL TUC) | (Sphere (cS1,TREC)) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL TUC
Euclid TUC is non empty strict Reflexive discerning V245() triangle MetrStruct
the carrier of (Euclid TUC) is non empty set
the carrier of (TUC,cS1,TREC) is set
TOUC is Element of the carrier of (Euclid TUC)
Sphere (TOUC,TREC) is Element of K6( the carrier of (Euclid TUC))
K6( the carrier of (Euclid TUC)) is non empty set
{TOUC} is non empty trivial Element of K6( the carrier of (Euclid TUC))
TUC is complex real ext-real set
(2,(0. (TOP-REAL 2)),TUC) is TopSpace-like V221() V222() SubSpace of TOP-REAL 2
Sphere ((0. (TOP-REAL 2)),TUC) is functional closed V258( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - (0. (TOP-REAL 2))).| = TUC } is set
(TOP-REAL 2) | (Sphere ((0. (TOP-REAL 2)),TUC)) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
- TUC is complex real ext-real set
Trectangle ((- TUC),TUC,(- TUC),TUC) is TopSpace-like V221() V222() SubSpace of TOP-REAL 2
closed_inside_of_rectangle ((- TUC),TUC,(- TUC),TUC) is functional Element of K6( the carrier of (TOP-REAL 2))
(TOP-REAL 2) | (closed_inside_of_rectangle ((- TUC),TUC,(- TUC),TUC)) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
the carrier of (2,(0. (TOP-REAL 2)),TUC) is set
the carrier of (Trectangle ((- TUC),TUC,(- TUC),TUC)) is set
TOUC is set
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : ( - TUC <= b₁ `1 & b<sub>1</sub> `1 <= TUC & - TUC <= b₁ `2 & b<sub>1</sub> `2 <= TUC ) } is set
TOUCm is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|.TOUCm.| is complex real ext-real non negative Element of REAL
TOUCm `2 is complex real ext-real Element of REAL
abs (TOUCm `2) is complex real ext-real Element of REAL
TOUCm `1 is complex real ext-real Element of REAL
abs (TOUCm `1) is complex real ext-real Element of REAL
TUC is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
cS1 is non empty complex real ext-real positive non negative set
(2,TUC,cS1) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
Sphere (TUC,cS1) is non empty functional closed closed compact V258( TOP-REAL 2) V258( TOP-REAL 2) being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - TUC).| = cS1 } is set
(TOP-REAL 2) | (Sphere (TUC,cS1)) is non empty strict TopSpace-like compact V221() V222() V266() SubSpace of TOP-REAL 2
the carrier of (2,TUC,cS1) is non empty set
the V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
the non empty complex real ext-real positive non negative set is non empty complex real ext-real positive non negative set
(2, the V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2), the non empty complex real ext-real positive non negative set ) is non empty strict TopSpace-like compact V221() V222() V266() pathwise_connected () SubSpace of TOP-REAL 2
Sphere ( the V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2), the non empty complex real ext-real positive non negative set ) is non empty functional closed closed compact V258( TOP-REAL 2) V258( TOP-REAL 2) being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - the V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)).| = the non empty complex real ext-real positive non negative set } is set
(TOP-REAL 2) | (Sphere ( the V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2), the non empty complex real ext-real positive non negative set )) is non empty strict TopSpace-like compact V221() V222() V266() SubSpace of TOP-REAL 2
TUC is non empty TopSpace-like compact V221() V222() V266() pathwise_connected () SubSpace of TOP-REAL 2
cS1 is non empty TopSpace-like compact V221() V222() V266() pathwise_connected () SubSpace of TOP-REAL 2
the carrier of TUC is non empty set
the topology of TUC is non empty Element of K6(K6( the carrier of TUC))
K6( the carrier of TUC) is non empty set
K6(K6( the carrier of TUC)) is non empty set
TopStruct(# the carrier of TUC, the topology of TUC #) is non empty strict TopSpace-like TopStruct
the carrier of cS1 is non empty set
the topology of cS1 is non empty Element of K6(K6( the carrier of cS1))
K6( the carrier of cS1) is non empty set
K6(K6( the carrier of cS1)) is non empty set
TopStruct(# the carrier of cS1, the topology of cS1 #) is non empty strict TopSpace-like TopStruct
the carrier of TopStruct(# the carrier of TUC, the topology of TUC #) is non empty set
TREC is non empty functional compact being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
(TOP-REAL 2) | TREC is non empty strict TopSpace-like compact V221() V222() V266() SubSpace of TOP-REAL 2
the carrier of ((TOP-REAL 2) | TREC) is non empty set
K7( the carrier of ((TOP-REAL 2) | R^2-unit_square), the carrier of ((TOP-REAL 2) | TREC)) is non empty set
K6(K7( the carrier of ((TOP-REAL 2) | R^2-unit_square), the carrier of ((TOP-REAL 2) | TREC))) is non empty set
TOUC is non empty V16() V19( the carrier of ((TOP-REAL 2) | R^2-unit_square)) V20( the carrier of ((TOP-REAL 2) | TREC)) Function-like total quasi_total Element of K6(K7( the carrier of ((TOP-REAL 2) | R^2-unit_square), the carrier of ((TOP-REAL 2) | TREC)))
TOUC /" is non empty V16() V19( the carrier of ((TOP-REAL 2) | TREC)) V20( the carrier of ((TOP-REAL 2) | R^2-unit_square)) Function-like total quasi_total Element of K6(K7( the carrier of ((TOP-REAL 2) | TREC), the carrier of ((TOP-REAL 2) | R^2-unit_square)))
K7( the carrier of ((TOP-REAL 2) | TREC), the carrier of ((TOP-REAL 2) | R^2-unit_square)) is non empty set
K6(K7( the carrier of ((TOP-REAL 2) | TREC), the carrier of ((TOP-REAL 2) | R^2-unit_square))) is non empty set
[#] TopStruct(# the carrier of TUC, the topology of TUC #) is non empty non proper open closed dense non boundary Element of K6( the carrier of TopStruct(# the carrier of TUC, the topology of TUC #))
K6( the carrier of TopStruct(# the carrier of TUC, the topology of TUC #)) is non empty set
the carrier of TopStruct(# the carrier of cS1, the topology of cS1 #) is non empty set
TOUCm is non empty functional compact being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
(TOP-REAL 2) | TOUCm is non empty strict TopSpace-like compact V221() V222() V266() SubSpace of TOP-REAL 2
the carrier of ((TOP-REAL 2) | TOUCm) is non empty set
K7( the carrier of ((TOP-REAL 2) | R^2-unit_square), the carrier of ((TOP-REAL 2) | TOUCm)) is non empty set
K6(K7( the carrier of ((TOP-REAL 2) | R^2-unit_square), the carrier of ((TOP-REAL 2) | TOUCm))) is non empty set
X is non empty V16() V19( the carrier of ((TOP-REAL 2) | R^2-unit_square)) V20( the carrier of ((TOP-REAL 2) | TOUCm)) Function-like total quasi_total Element of K6(K7( the carrier of ((TOP-REAL 2) | R^2-unit_square), the carrier of ((TOP-REAL 2) | TOUCm)))
[#] TopStruct(# the carrier of cS1, the topology of cS1 #) is non empty non proper open closed dense non boundary Element of K6( the carrier of TopStruct(# the carrier of cS1, the topology of cS1 #))
K6( the carrier of TopStruct(# the carrier of cS1, the topology of cS1 #)) is non empty set
K7( the carrier of TopStruct(# the carrier of TUC, the topology of TUC #), the carrier of TopStruct(# the carrier of cS1, the topology of cS1 #)) is non empty set
K6(K7( the carrier of TopStruct(# the carrier of TUC, the topology of TUC #), the carrier of TopStruct(# the carrier of cS1, the topology of cS1 #))) is non empty set
X * (TOUC /") is non empty V16() V19( the carrier of ((TOP-REAL 2) | TREC)) V20( the carrier of ((TOP-REAL 2) | TOUCm)) Function-like total quasi_total Element of K6(K7( the carrier of ((TOP-REAL 2) | TREC), the carrier of ((TOP-REAL 2) | TOUCm)))
K7( the carrier of ((TOP-REAL 2) | TREC), the carrier of ((TOP-REAL 2) | TOUCm)) is non empty set
K6(K7( the carrier of ((TOP-REAL 2) | TREC), the carrier of ((TOP-REAL 2) | TOUCm))) is non empty set
Xm is non empty V16() V19( the carrier of TopStruct(# the carrier of TUC, the topology of TUC #)) V20( the carrier of TopStruct(# the carrier of cS1, the topology of cS1 #)) Function-like total quasi_total Element of K6(K7( the carrier of TopStruct(# the carrier of TUC, the topology of TUC #), the carrier of TopStruct(# the carrier of cS1, the topology of cS1 #)))
TUC is ordinal natural complex real ext-real non negative integer set
TOP-REAL TUC is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
0. (TOP-REAL TUC) is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like zero Element of the carrier of (TOP-REAL TUC)
the carrier of (TOP-REAL TUC) is non empty functional set
the ZeroF of (TOP-REAL TUC) is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TUC)
(TUC,(0. (TOP-REAL TUC)),1) is TopSpace-like V221() V222() SubSpace of TOP-REAL TUC
Sphere ((0. (TOP-REAL TUC)),1) is functional closed Element of K6( the carrier of (TOP-REAL TUC))
K6( the carrier of (TOP-REAL TUC)) is non empty set
{ b₁ where b₁ is V16() Function-like V34(TUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TUC) : |.(b₁ - (0. (TOP-REAL TUC))).| = 1 } is set
(TOP-REAL TUC) | (Sphere ((0. (TOP-REAL TUC)),1)) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL TUC
(2) is TopSpace-like V221() V222() SubSpace of TOP-REAL 2
(2,(0. (TOP-REAL 2)),1) is non empty strict TopSpace-like compact V221() V222() V266() pathwise_connected () SubSpace of TOP-REAL 2
Sphere ((0. (TOP-REAL 2)),1) is non empty functional closed closed compact V258( TOP-REAL 2) V258( TOP-REAL 2) being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - (0. (TOP-REAL 2))).| = 1 } is set
(TOP-REAL 2) | (Sphere ((0. (TOP-REAL 2)),1)) is non empty strict TopSpace-like compact V221() V222() V266() SubSpace of TOP-REAL 2
the carrier of (2) is set
Sphere (|[0,0]|,1) is non empty functional closed closed compact V258( TOP-REAL 2) V258( TOP-REAL 2) being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - |[0,0]|).| = 1 } is set
TREC is non empty ordinal natural complex real ext-real positive non negative integer set
(TREC) is TopSpace-like V221() V222() SubSpace of TOP-REAL TREC
TOP-REAL TREC is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
0. (TOP-REAL TREC) is V16() Function-like V34(TREC) V51() V52() V53() FinSequence-like zero Element of the carrier of (TOP-REAL TREC)
the carrier of (TOP-REAL TREC) is non empty functional set
the ZeroF of (TOP-REAL TREC) is V16() Function-like V34(TREC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TREC)
(TREC,(0. (TOP-REAL TREC)),1) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TREC
Sphere ((0. (TOP-REAL TREC)),1) is non empty functional closed Element of K6( the carrier of (TOP-REAL TREC))
K6( the carrier of (TOP-REAL TREC)) is non empty set
{ b₁ where b₁ is V16() Function-like V34(TREC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TREC) : |.(b₁ - (0. (TOP-REAL TREC))).| = 1 } is set
(TOP-REAL TREC) | (Sphere ((0. (TOP-REAL TREC)),1)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TREC
TOUC is non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
TOP-REAL TOUC is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
the carrier of (TOP-REAL TOUC) is non empty functional set
(TOUC) is non empty TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
0. (TOP-REAL TOUC) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like zero Element of the carrier of (TOP-REAL TOUC)
the ZeroF of (TOP-REAL TOUC) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(TOUC,(0. (TOP-REAL TOUC)),1) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
Sphere ((0. (TOP-REAL TOUC)),1) is non empty functional closed V258( TOP-REAL TOUC) Element of K6( the carrier of (TOP-REAL TOUC))
K6( the carrier of (TOP-REAL TOUC)) is non empty set
{ b₁ where b₁ is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC) : |.(b₁ - (0. (TOP-REAL TOUC))).| = 1 } is set
(TOP-REAL TOUC) | (Sphere ((0. (TOP-REAL TOUC)),1)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
the carrier of (TOUC) is non empty set
TOUCm is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
|.TOUCm.| is complex real ext-real non negative Element of REAL
TREC is complex real ext-real non negative set
(TOUC,(0. (TOP-REAL TOUC)),TREC) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
Sphere ((0. (TOP-REAL TOUC)),TREC) is non empty functional closed V258( TOP-REAL TOUC) Element of K6( the carrier of (TOP-REAL TOUC))
{ b₁ where b₁ is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC) : |.(b₁ - (0. (TOP-REAL TOUC))).| = TREC } is set
(TOP-REAL TOUC) | (Sphere ((0. (TOP-REAL TOUC)),TREC)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
the carrier of (TOUC,(0. (TOP-REAL TOUC)),TREC) is non empty set
TREC is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
TREC `1 is complex real ext-real Element of REAL
TREC `2 is complex real ext-real Element of REAL
TOUC is complex real ext-real Element of REAL
TOUCm is complex real ext-real Element of REAL
<*TOUC,TOUCm*> is set
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
|.TREC.| is complex real ext-real non negative Element of REAL
|.TREC.| ^2 is complex real ext-real Element of REAL
|.TREC.| * |.TREC.| is complex real ext-real non negative set
TOUC ^2 is complex real ext-real Element of REAL
TOUC * TOUC is complex real ext-real set
TOUCm ^2 is complex real ext-real Element of REAL
TOUCm * TOUCm is complex real ext-real set
(TOUC ^2) + (TOUCm ^2) is complex real ext-real Element of REAL
- (TOUC ^2) is complex real ext-real Element of REAL
- 0 is empty trivial complex real ext-real non positive non negative Function-like functional integer V67() V68() V69() V70() V71() V72() V73() Element of REAL
(TOUCm ^2) - 1 is complex real ext-real Element of REAL
- (TOUCm ^2) is complex real ext-real Element of REAL
(TOUC ^2) - 1 is complex real ext-real Element of REAL
TREC is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
TREC `1 is complex real ext-real Element of REAL
TREC `2 is complex real ext-real Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
|.TREC.| is complex real ext-real non negative Element of REAL
|.TREC.| ^2 is complex real ext-real Element of REAL
|.TREC.| * |.TREC.| is complex real ext-real non negative set
(TREC `1) ^2 is complex real ext-real Element of REAL
(TREC `1) * (TREC `1) is complex real ext-real set
(TREC `2) ^2 is complex real ext-real Element of REAL
(TREC `2) * (TREC `2) is complex real ext-real set
((TREC `1) ^2) + ((TREC `2) ^2) is complex real ext-real Element of REAL
TREC is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
TREC `1 is complex real ext-real Element of REAL
TREC `2 is complex real ext-real Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
|.TREC.| is complex real ext-real non negative Element of REAL
|.TREC.| ^2 is complex real ext-real Element of REAL
|.TREC.| * |.TREC.| is complex real ext-real non negative set
(TREC `1) ^2 is complex real ext-real Element of REAL
(TREC `1) * (TREC `1) is complex real ext-real set
(TREC `2) ^2 is complex real ext-real Element of REAL
(TREC `2) * (TREC `2) is complex real ext-real set
((TREC `1) ^2) + ((TREC `2) ^2) is complex real ext-real Element of REAL
TREC is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
TREC `2 is complex real ext-real Element of REAL
TREC `1 is complex real ext-real Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
|.TREC.| is complex real ext-real non negative Element of REAL
|.TREC.| ^2 is complex real ext-real Element of REAL
|.TREC.| * |.TREC.| is complex real ext-real non negative set
(TREC `1) ^2 is complex real ext-real Element of REAL
(TREC `1) * (TREC `1) is complex real ext-real set
(TREC `2) ^2 is complex real ext-real Element of REAL
(TREC `2) * (TREC `2) is complex real ext-real set
((TREC `1) ^2) + ((TREC `2) ^2) is complex real ext-real Element of REAL
TREC is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
TREC `2 is complex real ext-real Element of REAL
TREC `1 is complex real ext-real Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
|.TREC.| is complex real ext-real non negative Element of REAL
|.TREC.| ^2 is complex real ext-real Element of REAL
|.TREC.| * |.TREC.| is complex real ext-real non negative set
(TREC `1) ^2 is complex real ext-real Element of REAL
(TREC `1) * (TREC `1) is complex real ext-real set
(TREC `2) ^2 is complex real ext-real Element of REAL
(TREC `2) * (TREC `2) is complex real ext-real set
((TREC `1) ^2) + ((TREC `2) ^2) is complex real ext-real Element of REAL
p0 is non empty complex real ext-real non positive negative set
p1 is non empty complex real ext-real positive non negative set
Trectangle (p0,p1,p0,p1) is non empty TopSpace-like V221() V222() SubSpace of TOP-REAL 2
closed_inside_of_rectangle (p0,p1,p0,p1) is functional Element of K6( the carrier of (TOP-REAL 2))
(TOP-REAL 2) | (closed_inside_of_rectangle (p0,p1,p0,p1)) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
Trectangle ((- 1),1,(- 1),1) is non empty TopSpace-like V221() V222() SubSpace of TOP-REAL 2
closed_inside_of_rectangle ((- 1),1,(- 1),1) is functional Element of K6( the carrier of (TOP-REAL 2))
(TOP-REAL 2) | (closed_inside_of_rectangle ((- 1),1,(- 1),1)) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
TOUC is non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
TOP-REAL TOUC is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
the carrier of (TOP-REAL TOUC) is non empty functional set
(TOUC) is non empty TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
0. (TOP-REAL TOUC) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like zero Element of the carrier of (TOP-REAL TOUC)
the ZeroF of (TOP-REAL TOUC) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(TOUC,(0. (TOP-REAL TOUC)),1) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
Sphere ((0. (TOP-REAL TOUC)),1) is non empty functional closed V258( TOP-REAL TOUC) Element of K6( the carrier of (TOP-REAL TOUC))
K6( the carrier of (TOP-REAL TOUC)) is non empty set
{ b₁ where b₁ is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC) : |.(b₁ - (0. (TOP-REAL TOUC))).| = 1 } is set
(TOP-REAL TOUC) | (Sphere ((0. (TOP-REAL TOUC)),1)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
the carrier of (TOUC) is non empty set
TOUCm is non empty complex real ext-real positive non negative set
X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(TOUC,X,TOUCm) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
Sphere (X,TOUCm) is non empty functional closed V258( TOP-REAL TOUC) Element of K6( the carrier of (TOP-REAL TOUC))
{ b₁ where b₁ is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC) : |.(b₁ - X).| = TOUCm } is set
(TOP-REAL TOUC) | (Sphere (X,TOUCm)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
the carrier of (TOUC,X,TOUCm) is non empty set
K7( the carrier of (TOUC), the carrier of (TOUC,X,TOUCm)) is non empty set
K6(K7( the carrier of (TOUC), the carrier of (TOUC,X,TOUCm))) is non empty set
Xm is non empty V16() V19( the carrier of (TOUC)) V20( the carrier of (TOUC,X,TOUCm)) Function-like total quasi_total Element of K6(K7( the carrier of (TOUC), the carrier of (TOUC,X,TOUCm)))
A1 is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
TOUCm * A1 is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(A1,TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * A1) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
the U6 of (TOP-REAL TOUC) is non empty V16() V19(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)))
K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)) is non empty set
K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC))) is non empty set
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * A1),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * A1),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) is non empty set
A1 is non empty V16() V19( the carrier of (TOP-REAL TOUC)) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)))
1 / TOUCm is non empty complex real ext-real positive non negative Element of REAL
C is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
C - X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
- X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K543(X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL TOUC),C,(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
the U6 of (TOP-REAL TOUC) is non empty V16() V19(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)))
K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC))) is non empty set
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),C,(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(C,(- X)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(C,X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(1 / TOUCm) * (C - X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547((C - X),(1 / TOUCm)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
C is non empty V16() V19( the carrier of (TOP-REAL TOUC)) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)))
(TOP-REAL TOUC) --> X is non empty V16() V19( the carrier of (TOP-REAL TOUC)) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total continuous Function-yielding V312() Element of K6(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)))
the carrier of (TOP-REAL TOUC) --> X is non empty V16() V19( the carrier of (TOP-REAL TOUC)) V20( the carrier of (TOP-REAL TOUC)) Function-like constant total quasi_total Function-yielding V312() Element of K6(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)))
id (TOP-REAL TOUC) is non empty V16() V19( the carrier of (TOP-REAL TOUC)) V20( the carrier of (TOP-REAL TOUC)) Function-like one-to-one total quasi_total onto bijective continuous being_homeomorphism open Function-yielding V312() Element of K6(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)))
id the carrier of (TOP-REAL TOUC) is non empty V16() V19( the carrier of (TOP-REAL TOUC)) V20( the carrier of (TOP-REAL TOUC)) Function-like one-to-one total quasi_total Function-yielding V312() Element of K6(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)))
dom C is functional Element of K6( the carrier of (TOP-REAL TOUC))
C | (Sphere (X,TOUCm)) is V16() V19( the carrier of (TOP-REAL TOUC)) V20( the carrier of (TOP-REAL TOUC)) Function-like Function-yielding V312() Element of K6(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)))
dom (C | (Sphere (X,TOUCm))) is functional Element of K6( the carrier of (TOP-REAL TOUC))
- (1 / TOUCm) is non empty complex real ext-real non positive negative Element of REAL
Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
C . Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(id (TOP-REAL TOUC)) . Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(1 / TOUCm) * ((id (TOP-REAL TOUC)) . Q) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(((id (TOP-REAL TOUC)) . Q),(1 / TOUCm)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((TOP-REAL TOUC) --> X) . Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(- (1 / TOUCm)) * (((TOP-REAL TOUC) --> X) . Q) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547((((TOP-REAL TOUC) --> X) . Q),(- (1 / TOUCm))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((1 / TOUCm) * ((id (TOP-REAL TOUC)) . Q)) + ((- (1 / TOUCm)) * (((TOP-REAL TOUC) --> X) . Q)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
the U6 of (TOP-REAL TOUC) is non empty V16() V19(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)))
K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC))) is non empty set
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),((1 / TOUCm) * ((id (TOP-REAL TOUC)) . Q)),((- (1 / TOUCm)) * (((TOP-REAL TOUC) --> X) . Q))) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(((1 / TOUCm) * ((id (TOP-REAL TOUC)) . Q)),((- (1 / TOUCm)) * (((TOP-REAL TOUC) --> X) . Q))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(1 / TOUCm) * Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(Q,(1 / TOUCm)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((1 / TOUCm) * Q) + ((- (1 / TOUCm)) * (((TOP-REAL TOUC) --> X) . Q)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),((1 / TOUCm) * Q),((- (1 / TOUCm)) * (((TOP-REAL TOUC) --> X) . Q))) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(((1 / TOUCm) * Q),((- (1 / TOUCm)) * (((TOP-REAL TOUC) --> X) . Q))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(- (1 / TOUCm)) * X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(X,(- (1 / TOUCm))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((1 / TOUCm) * Q) + ((- (1 / TOUCm)) * X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),((1 / TOUCm) * Q),((- (1 / TOUCm)) * X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(((1 / TOUCm) * Q),((- (1 / TOUCm)) * X)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(1 / TOUCm) * X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(X,(1 / TOUCm)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((1 / TOUCm) * Q) - ((1 / TOUCm) * X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
- ((1 / TOUCm) * X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K543(((1 / TOUCm) * X)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL TOUC),((1 / TOUCm) * Q),(- ((1 / TOUCm) * X))) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),((1 / TOUCm) * Q),(- ((1 / TOUCm) * X))) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(((1 / TOUCm) * Q),(- ((1 / TOUCm) * X))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(((1 / TOUCm) * Q),((1 / TOUCm) * X)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
Q - X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
- X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K543(X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL TOUC),Q,(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),Q,(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(Q,(- X)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(Q,X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(1 / TOUCm) * (Q - X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547((Q - X),(1 / TOUCm)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
dom Xm is Element of K6( the carrier of (TOUC))
K6( the carrier of (TOUC)) is non empty set
[#] (TOUC) is non empty non proper open closed dense non boundary Element of K6( the carrier of (TOUC))
rng Xm is Element of K6( the carrier of (TOUC,X,TOUCm))
K6( the carrier of (TOUC,X,TOUCm)) is non empty set
[#] (TOUC,X,TOUCm) is non empty non proper open closed dense non boundary Element of K6( the carrier of (TOUC,X,TOUCm))
Xm /" is non empty V16() V19( the carrier of (TOUC,X,TOUCm)) V20( the carrier of (TOUC)) Function-like total quasi_total Element of K6(K7( the carrier of (TOUC,X,TOUCm), the carrier of (TOUC)))
K7( the carrier of (TOUC,X,TOUCm), the carrier of (TOUC)) is non empty set
K6(K7( the carrier of (TOUC,X,TOUCm), the carrier of (TOUC))) is non empty set
Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
A1 . Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(id (TOP-REAL TOUC)) . Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
TOUCm * ((id (TOP-REAL TOUC)) . Q) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(((id (TOP-REAL TOUC)) . Q),TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((TOP-REAL TOUC) --> X) . Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
1 * (((TOP-REAL TOUC) --> X) . Q) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547((((TOP-REAL TOUC) --> X) . Q),1) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * ((id (TOP-REAL TOUC)) . Q)) + (1 * (((TOP-REAL TOUC) --> X) . Q)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
the U6 of (TOP-REAL TOUC) is non empty V16() V19(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)))
K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC))) is non empty set
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * ((id (TOP-REAL TOUC)) . Q)),(1 * (((TOP-REAL TOUC) --> X) . Q))) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * ((id (TOP-REAL TOUC)) . Q)),(1 * (((TOP-REAL TOUC) --> X) . Q))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * ((id (TOP-REAL TOUC)) . Q)) + (((TOP-REAL TOUC) --> X) . Q) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * ((id (TOP-REAL TOUC)) . Q)),(((TOP-REAL TOUC) --> X) . Q)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * ((id (TOP-REAL TOUC)) . Q)),(((TOP-REAL TOUC) --> X) . Q)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
TOUCm * Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(Q,TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * Q) + (((TOP-REAL TOUC) --> X) . Q) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * Q),(((TOP-REAL TOUC) --> X) . Q)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * Q),(((TOP-REAL TOUC) --> X) . Q)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * Q) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * Q),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * Q),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
A1 | (Sphere ((0. (TOP-REAL TOUC)),1)) is V16() V19( the carrier of (TOP-REAL TOUC)) V20( the carrier of (TOP-REAL TOUC)) Function-like Function-yielding V312() Element of K6(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)))
Q is set
Xm . Q is set
(A1 | (Sphere ((0. (TOP-REAL TOUC)),1))) . Q is V16() Function-like set
E is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
TOUCm * E is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(E,TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * E) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
the U6 of (TOP-REAL TOUC) is non empty V16() V19(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)))
K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC))) is non empty set
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * E),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * E),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
A1 . E is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(1 / TOUCm) * TOUCm is non empty complex real ext-real positive non negative Element of REAL
Q is set
E is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
E - X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
- X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K543(X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL TOUC),E,(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
the U6 of (TOP-REAL TOUC) is non empty V16() V19(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)))
K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC))) is non empty set
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),E,(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(E,(- X)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(E,X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(1 / TOUCm) * (E - X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547((E - X),(1 / TOUCm)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((1 / TOUCm) * (E - X)) - (0. (TOP-REAL TOUC)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
- (0. (TOP-REAL TOUC)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K543((0. (TOP-REAL TOUC))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL TOUC),((1 / TOUCm) * (E - X)),(- (0. (TOP-REAL TOUC)))) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),((1 / TOUCm) * (E - X)),(- (0. (TOP-REAL TOUC)))) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(((1 / TOUCm) * (E - X)),(- (0. (TOP-REAL TOUC)))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(((1 / TOUCm) * (E - X)),(0. (TOP-REAL TOUC))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(((1 / TOUCm) * (E - X)) - (0. (TOP-REAL TOUC))).| is complex real ext-real non negative Element of REAL
|.((1 / TOUCm) * (E - X)).| is complex real ext-real non negative Element of REAL
abs (1 / TOUCm) is complex real ext-real Element of REAL
|.(E - X).| is complex real ext-real non negative Element of REAL
(abs (1 / TOUCm)) * |.(E - X).| is complex real ext-real Element of REAL
(1 / TOUCm) * |.(E - X).| is complex real ext-real non negative Element of REAL
Xm . ((1 / TOUCm) * (E - X)) is set
TOUCm * ((1 / TOUCm) * (E - X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(((1 / TOUCm) * (E - X)),TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * ((1 / TOUCm) * (E - X))) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * ((1 / TOUCm) * (E - X))),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * ((1 / TOUCm) * (E - X))),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
TOUCm * (1 / TOUCm) is non empty complex real ext-real positive non negative Element of REAL
(TOUCm * (1 / TOUCm)) * (E - X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547((E - X),(TOUCm * (1 / TOUCm))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((TOUCm * (1 / TOUCm)) * (E - X)) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),((TOUCm * (1 / TOUCm)) * (E - X)),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(((TOUCm * (1 / TOUCm)) * (E - X)),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(E - X) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(E - X),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((E - X),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
Q is set
K121(Xm) is set
E is set
Xm . Q is set
Xm . E is set
Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
Xm . Q is set
TOUCm * Q is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(Q,TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * Q) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
the U6 of (TOP-REAL TOUC) is non empty V16() V19(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)))
K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC))) is non empty set
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * Q),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * Q),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
j is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
Xm . j is set
TOUCm * j is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(j,TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * j) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * j),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * j),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((TOUCm * Q) + X) - X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
- X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K543(X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL TOUC),((TOUCm * Q) + X),(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),((TOUCm * Q) + X),(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(((TOUCm * Q) + X),(- X)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(((TOUCm * Q) + X),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
dom (Xm /") is Element of K6( the carrier of (TOUC,X,TOUCm))
E is set
(Xm /") . E is set
(C | (Sphere (X,TOUCm))) . E is V16() Function-like set
j is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
j - X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
- X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K543(X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL TOUC),j,(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
the U6 of (TOP-REAL TOUC) is non empty V16() V19(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)))
K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC))) is non empty set
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),j,(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(j,(- X)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(j,X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(1 / TOUCm) * (j - X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547((j - X),(1 / TOUCm)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((1 / TOUCm) * (j - X)) - (0. (TOP-REAL TOUC)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
- (0. (TOP-REAL TOUC)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K543((0. (TOP-REAL TOUC))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL TOUC),((1 / TOUCm) * (j - X)),(- (0. (TOP-REAL TOUC)))) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),((1 / TOUCm) * (j - X)),(- (0. (TOP-REAL TOUC)))) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(((1 / TOUCm) * (j - X)),(- (0. (TOP-REAL TOUC)))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(((1 / TOUCm) * (j - X)),(0. (TOP-REAL TOUC))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(((1 / TOUCm) * (j - X)) - (0. (TOP-REAL TOUC))).| is complex real ext-real non negative Element of REAL
|.((1 / TOUCm) * (j - X)).| is complex real ext-real non negative Element of REAL
abs (1 / TOUCm) is complex real ext-real Element of REAL
|.(j - X).| is complex real ext-real non negative Element of REAL
(abs (1 / TOUCm)) * |.(j - X).| is complex real ext-real Element of REAL
(1 / TOUCm) * |.(j - X).| is complex real ext-real non negative Element of REAL
Xm . ((1 / TOUCm) * (j - X)) is set
TOUCm * ((1 / TOUCm) * (j - X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(((1 / TOUCm) * (j - X)),TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * ((1 / TOUCm) * (j - X))) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * ((1 / TOUCm) * (j - X))),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * ((1 / TOUCm) * (j - X))),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
TOUCm * (1 / TOUCm) is non empty complex real ext-real positive non negative Element of REAL
(TOUCm * (1 / TOUCm)) * (j - X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547((j - X),(TOUCm * (1 / TOUCm))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((TOUCm * (1 / TOUCm)) * (j - X)) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),((TOUCm * (1 / TOUCm)) * (j - X)),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(((TOUCm * (1 / TOUCm)) * (j - X)),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(j - X) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(j - X),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((j - X),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
Q is V16() Function-like set
Q " is V16() Function-like set
(Q ") . E is set
C . j is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
dom A1 is functional Element of K6( the carrier of (TOP-REAL TOUC))
dom (A1 | (Sphere ((0. (TOP-REAL TOUC)),1))) is functional Element of K6( the carrier of (TOP-REAL TOUC))
TOUC is non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
TOP-REAL TOUC is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
the carrier of (TOP-REAL TOUC) is non empty functional set
(TOUC) is non empty TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
0. (TOP-REAL TOUC) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like zero Element of the carrier of (TOP-REAL TOUC)
the ZeroF of (TOP-REAL TOUC) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(TOUC,(0. (TOP-REAL TOUC)),1) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
Sphere ((0. (TOP-REAL TOUC)),1) is non empty functional closed V258( TOP-REAL TOUC) Element of K6( the carrier of (TOP-REAL TOUC))
K6( the carrier of (TOP-REAL TOUC)) is non empty set
{ b₁ where b₁ is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC) : |.(b₁ - (0. (TOP-REAL TOUC))).| = 1 } is set
(TOP-REAL TOUC) | (Sphere ((0. (TOP-REAL TOUC)),1)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
TOUCm is non empty complex real ext-real positive non negative set
X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(TOUC,X,TOUCm) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
Sphere (X,TOUCm) is non empty functional closed V258( TOP-REAL TOUC) Element of K6( the carrier of (TOP-REAL TOUC))
{ b₁ where b₁ is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC) : |.(b₁ - X).| = TOUCm } is set
(TOP-REAL TOUC) | (Sphere (X,TOUCm)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
the carrier of (TOUC) is non empty set
the carrier of (TOUC,X,TOUCm) is non empty set
Ym is V16() Function-like FinSequence-like Element of the carrier of (TOUC)
A1 is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
TOUCm * A1 is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(A1,TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * A1) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
the U6 of (TOP-REAL TOUC) is non empty V16() V19(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)))
K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)) is non empty set
K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC))) is non empty set
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * A1),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * A1),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
((TOUCm * A1) + X) - X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
- X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K543(X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL TOUC),((TOUCm * A1) + X),(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),((TOUCm * A1) + X),(- X)) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541(((TOUCm * A1) + X),(- X)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(((TOUCm * A1) + X),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(((TOUCm * A1) + X) - X).| is complex real ext-real non negative Element of REAL
|.(TOUCm * A1).| is complex real ext-real non negative Element of REAL
abs TOUCm is complex real ext-real Element of REAL
|.A1.| is complex real ext-real non negative Element of REAL
(abs TOUCm) * |.A1.| is complex real ext-real Element of REAL
TOUCm * |.A1.| is complex real ext-real non negative Element of REAL
TOUCm * 1 is non empty complex real ext-real positive non negative Element of REAL
Cm is V16() Function-like FinSequence-like Element of the carrier of (TOUC,X,TOUCm)
K7( the carrier of (TOUC), the carrier of (TOUC,X,TOUCm)) is non empty set
K6(K7( the carrier of (TOUC), the carrier of (TOUC,X,TOUCm))) is non empty set
Ym is non empty V16() V19( the carrier of (TOUC)) V20( the carrier of (TOUC,X,TOUCm)) Function-like total quasi_total Element of K6(K7( the carrier of (TOUC), the carrier of (TOUC,X,TOUCm)))
A1 is V16() Function-like FinSequence-like Element of the carrier of (TOUC)
Ym . A1 is V16() Function-like FinSequence-like Element of the carrier of (TOUC,X,TOUCm)
C is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
TOUCm * C is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(C,TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * C) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
the U6 of (TOP-REAL TOUC) is non empty V16() V19(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC))) V20( the carrier of (TOP-REAL TOUC)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)))
K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)) is non empty set
K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL TOUC), the carrier of (TOP-REAL TOUC)), the carrier of (TOP-REAL TOUC))) is non empty set
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * C),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * C),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
Cm is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
TOUCm * Cm is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K547(Cm,TOUCm) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(TOUCm * Cm) + X is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K414( the carrier of (TOP-REAL TOUC), the U6 of (TOP-REAL TOUC),(TOUCm * Cm),X) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
K541((TOUCm * Cm),X) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
TOUC is non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
TOP-REAL TOUC is non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() L19()
the carrier of (TOP-REAL TOUC) is non empty functional set
TOUCm is non empty complex real ext-real positive non negative set
X is non empty complex real ext-real positive non negative set
Xm is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(TOUC,Xm,TOUCm) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
Sphere (Xm,TOUCm) is non empty functional closed V258( TOP-REAL TOUC) Element of K6( the carrier of (TOP-REAL TOUC))
K6( the carrier of (TOP-REAL TOUC)) is non empty set
{ b₁ where b₁ is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC) : |.(b₁ - Xm).| = TOUCm } is set
(TOP-REAL TOUC) | (Sphere (Xm,TOUCm)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
Y is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(TOUC,Y,X) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
Sphere (Y,X) is non empty functional closed V258( TOP-REAL TOUC) Element of K6( the carrier of (TOP-REAL TOUC))
{ b₁ where b₁ is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC) : |.(b₁ - Y).| = X } is set
(TOP-REAL TOUC) | (Sphere (Y,X)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
(TOUC) is non empty TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
0. (TOP-REAL TOUC) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like zero Element of the carrier of (TOP-REAL TOUC)
the ZeroF of (TOP-REAL TOUC) is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC)
(TOUC,(0. (TOP-REAL TOUC)),1) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
Sphere ((0. (TOP-REAL TOUC)),1) is non empty functional closed V258( TOP-REAL TOUC) Element of K6( the carrier of (TOP-REAL TOUC))
{ b₁ where b₁ is V16() Function-like V34(TOUC) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL TOUC) : |.(b₁ - (0. (TOP-REAL TOUC))).| = 1 } is set
(TOP-REAL TOUC) | (Sphere ((0. (TOP-REAL TOUC)),1)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL TOUC
TOUC is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
TOUCm is complex real ext-real non negative set
(2,TOUC,TOUCm) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
Sphere (TOUC,TOUCm) is non empty functional closed V258( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - TOUC).| = TOUCm } is set
(TOP-REAL 2) | (Sphere (TOUC,TOUCm)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
X is non empty complex real ext-real positive non negative set
(2,TOUC,X) is non empty strict TopSpace-like compact V221() V222() V266() pathwise_connected () SubSpace of TOP-REAL 2
Sphere (TOUC,X) is non empty functional closed closed compact V258( TOP-REAL 2) V258( TOP-REAL 2) being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - TOUC).| = X } is set
(TOP-REAL 2) | (Sphere (TOUC,X)) is non empty strict TopSpace-like compact V221() V222() V266() SubSpace of TOP-REAL 2
X is empty trivial complex real ext-real non positive non negative Function-like functional V67() V68() V69() V70() V71() V72() V73() set
(2,TOUC,X) is non empty trivial V89() 1 -element strict TopSpace-like V221() V222() pathwise_connected SubSpace of TOP-REAL 2
Sphere (TOUC,X) is non empty functional closed V258( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - TOUC).| = X } is set
(TOP-REAL 2) | (Sphere (TOUC,X)) is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
|[1,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[1,0]| `2 is complex real ext-real Element of REAL
|[1,0]| `1 is complex real ext-real Element of REAL
1 + 0 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
0 + 0 is empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V67() V68() V69() V70() V71() V72() V73() Element of REAL
|[(1 + 0),(0 + 0)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[(1 + 0),(0 + 0)]| - |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
- |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K543(|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL 2),|[(1 + 0),(0 + 0)]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
the U6 of (TOP-REAL 2) is non empty V16() V19(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2))) V20( the carrier of (TOP-REAL 2)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2)))
K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)) is non empty set
K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2))) is non empty set
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[(1 + 0),(0 + 0)]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[(1 + 0),(0 + 0)]|,(- |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(|[(1 + 0),(0 + 0)]|,|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(|[(1 + 0),(0 + 0)]| - |[0,0]|).| is complex real ext-real non negative Element of REAL
|[1,0]| + |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[1,0]|,|[0,0]|) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[1,0]|,|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(|[1,0]| + |[0,0]|) - |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K394((TOP-REAL 2),(|[1,0]| + |[0,0]|),(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),(|[1,0]| + |[0,0]|),(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541((|[1,0]| + |[0,0]|),(- |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545((|[1,0]| + |[0,0]|),|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.((|[1,0]| + |[0,0]|) - |[0,0]|).| is complex real ext-real non negative Element of REAL
|[0,0]| - |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K394((TOP-REAL 2),|[0,0]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[0,0]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[0,0]|,(- |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(|[0,0]|,|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|[1,0]| + (|[0,0]| - |[0,0]|) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[1,0]|,(|[0,0]| - |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[1,0]|,(|[0,0]| - |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(|[1,0]| + (|[0,0]| - |[0,0]|)).| is complex real ext-real non negative Element of REAL
|[1,0]| + (0. (TOP-REAL 2)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[1,0]|,(0. (TOP-REAL 2))) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[1,0]|,(0. (TOP-REAL 2))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(|[1,0]| + (0. (TOP-REAL 2))).| is complex real ext-real non negative Element of REAL
|.|[1,0]|.| is complex real ext-real non negative Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
0 ^2 is complex real ext-real Element of REAL
0 * 0 is empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V67() V68() V69() V70() V71() V72() V73() set
(1 ^2) + (0 ^2) is complex real ext-real Element of REAL
sqrt ((1 ^2) + (0 ^2)) is complex real ext-real Element of REAL
|[(- 1),0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[(- 1),0]| `2 is complex real ext-real Element of REAL
|[(- 1),0]| `1 is complex real ext-real Element of REAL
(- 1) + 0 is non empty complex real ext-real non positive negative integer Element of REAL
0 + 0 is empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V67() V68() V69() V70() V71() V72() V73() Element of REAL
|[((- 1) + 0),(0 + 0)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[((- 1) + 0),(0 + 0)]| - |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
- |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K543(|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL 2),|[((- 1) + 0),(0 + 0)]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
the U6 of (TOP-REAL 2) is non empty V16() V19(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2))) V20( the carrier of (TOP-REAL 2)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2)))
K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)) is non empty set
K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2))) is non empty set
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[((- 1) + 0),(0 + 0)]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[((- 1) + 0),(0 + 0)]|,(- |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(|[((- 1) + 0),(0 + 0)]|,|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(|[((- 1) + 0),(0 + 0)]| - |[0,0]|).| is complex real ext-real non negative Element of REAL
|[(- 1),0]| + |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[(- 1),0]|,|[0,0]|) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[(- 1),0]|,|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(|[(- 1),0]| + |[0,0]|) - |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K394((TOP-REAL 2),(|[(- 1),0]| + |[0,0]|),(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),(|[(- 1),0]| + |[0,0]|),(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541((|[(- 1),0]| + |[0,0]|),(- |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545((|[(- 1),0]| + |[0,0]|),|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.((|[(- 1),0]| + |[0,0]|) - |[0,0]|).| is complex real ext-real non negative Element of REAL
|[0,0]| - |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K394((TOP-REAL 2),|[0,0]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[0,0]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[0,0]|,(- |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(|[0,0]|,|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|[(- 1),0]| + (|[0,0]| - |[0,0]|) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[(- 1),0]|,(|[0,0]| - |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[(- 1),0]|,(|[0,0]| - |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(|[(- 1),0]| + (|[0,0]| - |[0,0]|)).| is complex real ext-real non negative Element of REAL
|[(- 1),0]| + (0. (TOP-REAL 2)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[(- 1),0]|,(0. (TOP-REAL 2))) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[(- 1),0]|,(0. (TOP-REAL 2))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(|[(- 1),0]| + (0. (TOP-REAL 2))).| is complex real ext-real non negative Element of REAL
|.|[(- 1),0]|.| is complex real ext-real non negative Element of REAL
(- 1) ^2 is complex real ext-real Element of REAL
(- 1) * (- 1) is non empty complex real ext-real positive non negative integer set
0 ^2 is complex real ext-real Element of REAL
0 * 0 is empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V67() V68() V69() V70() V71() V72() V73() set
((- 1) ^2) + (0 ^2) is complex real ext-real Element of REAL
sqrt (((- 1) ^2) + (0 ^2)) is complex real ext-real Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
(1 ^2) + (0 ^2) is complex real ext-real Element of REAL
sqrt ((1 ^2) + (0 ^2)) is complex real ext-real Element of REAL
() is V16() Function-like FinSequence-like Element of the carrier of (2)
() is V16() Function-like FinSequence-like Element of the carrier of (2)
TOUC is V16() Function-like FinSequence-like Element of the carrier of (2)
{TOUC} is non empty trivial functional compact Element of K6( the carrier of (2))
K6( the carrier of (2)) is non empty set
the carrier of (2) \ {TOUC} is Element of K6( the carrier of (2))
TOUCm is Element of K6( the carrier of (2))
(2) | TOUCm is strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of ((2) | TOUCm) is set
TOUCm is strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of TOUCm is set
X is strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of X is set
TOUC is V16() Function-like FinSequence-like Element of the carrier of (2)
(TOUC) is strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of (TOUC) is set
{TOUC} is non empty trivial functional compact Element of K6( the carrier of (2))
the carrier of (2) \ {TOUC} is Element of K6( the carrier of (2))
|[0,1]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[0,1]| `1 is complex real ext-real Element of REAL
|[0,1]| `2 is complex real ext-real Element of REAL
0 + 0 is empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V67() V68() V69() V70() V71() V72() V73() Element of REAL
1 + 0 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
|[(0 + 0),(1 + 0)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[(0 + 0),(1 + 0)]| - |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
- |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K543(|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL 2),|[(0 + 0),(1 + 0)]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
the U6 of (TOP-REAL 2) is non empty V16() V19(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2))) V20( the carrier of (TOP-REAL 2)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2)))
K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)) is non empty set
K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2))) is non empty set
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[(0 + 0),(1 + 0)]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[(0 + 0),(1 + 0)]|,(- |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(|[(0 + 0),(1 + 0)]|,|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(|[(0 + 0),(1 + 0)]| - |[0,0]|).| is complex real ext-real non negative Element of REAL
|[0,1]| + |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[0,1]|,|[0,0]|) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[0,1]|,|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
(|[0,1]| + |[0,0]|) - |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K394((TOP-REAL 2),(|[0,1]| + |[0,0]|),(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),(|[0,1]| + |[0,0]|),(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541((|[0,1]| + |[0,0]|),(- |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545((|[0,1]| + |[0,0]|),|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.((|[0,1]| + |[0,0]|) - |[0,0]|).| is complex real ext-real non negative Element of REAL
|[0,0]| - |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K394((TOP-REAL 2),|[0,0]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[0,0]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[0,0]|,(- |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(|[0,0]|,|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|[0,1]| + (|[0,0]| - |[0,0]|) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[0,1]|,(|[0,0]| - |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[0,1]|,(|[0,0]| - |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(|[0,1]| + (|[0,0]| - |[0,0]|)).| is complex real ext-real non negative Element of REAL
|[0,1]| + (0. (TOP-REAL 2)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[0,1]|,(0. (TOP-REAL 2))) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[0,1]|,(0. (TOP-REAL 2))) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(|[0,1]| + (0. (TOP-REAL 2))).| is complex real ext-real non negative Element of REAL
|.|[0,1]|.| is complex real ext-real non negative Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
0 ^2 is complex real ext-real Element of REAL
0 * 0 is empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V67() V68() V69() V70() V71() V72() V73() set
(1 ^2) + (0 ^2) is complex real ext-real Element of REAL
sqrt ((1 ^2) + (0 ^2)) is complex real ext-real Element of REAL
Xm is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Xm `1 is complex real ext-real Element of REAL
TOUC is V16() Function-like FinSequence-like Element of the carrier of (2)
(TOUC) is non empty strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of (TOUC) is non empty set
{TOUC} is non empty trivial functional compact Element of K6( the carrier of (2))
the carrier of (2) \ {TOUC} is Element of K6( the carrier of (2))
[#] (2) is non empty non proper open closed dense non boundary Element of K6( the carrier of (2))
TOUC is V16() Function-like FinSequence-like Element of the carrier of (2)
(TOUC) is non empty strict TopSpace-like V221() V222() SubSpace of (2)
{TOUC} is non empty trivial functional compact Element of K6( the carrier of (2))
([#] (2)) \ {TOUC} is Element of K6( the carrier of (2))
(2) | (([#] (2)) \ {TOUC}) is strict TopSpace-like V221() V222() SubSpace of (2)
[#] (TOUC) is non empty non proper open closed dense non boundary Element of K6( the carrier of (TOUC))
the carrier of (TOUC) is non empty set
K6( the carrier of (TOUC)) is non empty set
TOUC is V16() Function-like FinSequence-like Element of the carrier of (2)
TOUCm is V16() Function-like FinSequence-like Element of the carrier of (2)
(TOUC) is non empty strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of (TOUC) is non empty set
{TOUC} is non empty trivial functional compact Element of K6( the carrier of (2))
the carrier of (2) \ {TOUC} is Element of K6( the carrier of (2))
(()) is non empty strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of (()) is non empty set
TOUC is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
TOUC `2 is complex real ext-real Element of REAL
TOUC `1 is complex real ext-real Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
|.TOUC.| is complex real ext-real non negative Element of REAL
|.TOUC.| ^2 is complex real ext-real Element of REAL
|.TOUC.| * |.TOUC.| is complex real ext-real non negative set
(TOUC `1) ^2 is complex real ext-real Element of REAL
(TOUC `1) * (TOUC `1) is complex real ext-real set
(TOUC `2) ^2 is complex real ext-real Element of REAL
(TOUC `2) * (TOUC `2) is complex real ext-real set
((TOUC `1) ^2) + ((TOUC `2) ^2) is complex real ext-real Element of REAL
(()) is non empty strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of (()) is non empty set
TOUC is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
TOUC `2 is complex real ext-real Element of REAL
TOUC `1 is complex real ext-real Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
|.TOUC.| is complex real ext-real non negative Element of REAL
|.TOUC.| ^2 is complex real ext-real Element of REAL
|.TOUC.| * |.TOUC.| is complex real ext-real non negative set
(TOUC `1) ^2 is complex real ext-real Element of REAL
(TOUC `1) * (TOUC `1) is complex real ext-real set
(TOUC `2) ^2 is complex real ext-real Element of REAL
(TOUC `2) * (TOUC `2) is complex real ext-real set
((TOUC `1) ^2) + ((TOUC `2) ^2) is complex real ext-real Element of REAL
0 + p1 is non empty complex real ext-real positive non negative Element of REAL
].0,(0 + p1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].0,(0 + p1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].0,(0 + p1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].0,(0 + p1).[)) is non empty V67() V68() V69() set
(1 / 2) + p1 is non empty complex real ext-real positive non negative Element of REAL
].(1 / 2),((1 / 2) + p1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].(1 / 2),((1 / 2) + p1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(1 / 2),((1 / 2) + p1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)) is non empty V67() V68() V69() set
[#] (()) is non empty non proper open closed dense non boundary Element of K6( the carrier of (()))
K6( the carrier of (())) is non empty set
[#] (()) is non empty non proper open closed dense non boundary Element of K6( the carrier of (()))
K6( the carrier of (())) is non empty set
A1 is V16() Function-like FinSequence-like Element of the carrier of (2)
(A1) is non empty strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of (A1) is non empty set
C is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
C `1 is complex real ext-real Element of REAL
C `2 is complex real ext-real Element of REAL
A1 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A1 `1 is complex real ext-real Element of REAL
{()} is non empty trivial functional compact Element of K6( the carrier of (2))
the carrier of (2) \ {()} is Element of K6( the carrier of (2))
A1 `2 is complex real ext-real Element of REAL
|[(A1 `1),(A1 `2)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A1 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A1 `1 is complex real ext-real Element of REAL
{()} is non empty trivial functional compact Element of K6( the carrier of (2))
the carrier of (2) \ {()} is Element of K6( the carrier of (2))
A1 `2 is complex real ext-real Element of REAL
|[(A1 `1),(A1 `2)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A1 is V16() Function-like FinSequence-like Element of the carrier of (2)
(A1) is non empty strict TopSpace-like V221() V222() SubSpace of (2)
C is Element of K6( the carrier of (2))
the carrier of (A1) is non empty set
C ` is Element of K6( the carrier of (2))
the carrier of (2) \ C is set
{A1} is non empty trivial functional compact Element of K6( the carrier of (2))
the carrier of (2) \ {A1} is Element of K6( the carrier of (2))
the carrier of (2) \ ( the carrier of (2) \ {A1}) is Element of K6( the carrier of (2))
the carrier of (2) /\ {A1} is Element of K6( the carrier of (2))
Sphere ((0. (TOP-REAL 2)),p1) is non empty functional closed closed compact V258( TOP-REAL 2) V258( TOP-REAL 2) being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - (0. (TOP-REAL 2))).| = p1 } is set
C is V16() Function-like FinSequence-like Element of the carrier of (2)
(C) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
{C} is non empty trivial functional compact Element of K6( the carrier of (2))
(Sphere ((0. (TOP-REAL 2)),p1)) \ {C} is functional Element of K6( the carrier of (TOP-REAL 2))
the carrier of (2,(0. (TOP-REAL 2)),1) is non empty set
K6( the carrier of (2,(0. (TOP-REAL 2)),1)) is non empty set
Q is Element of K6( the carrier of (2,(0. (TOP-REAL 2)),1))
(2,(0. (TOP-REAL 2)),1) | Q is strict TopSpace-like V221() V222() SubSpace of (2,(0. (TOP-REAL 2)),1)
A is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
{A} is non empty trivial functional compact Element of K6( the carrier of (TOP-REAL 2))
(Sphere ((0. (TOP-REAL 2)),p1)) \ {A} is functional Element of K6( the carrier of (TOP-REAL 2))
(TOP-REAL 2) | ((Sphere ((0. (TOP-REAL 2)),p1)) \ {A}) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
Sphere ((0. (TOP-REAL 2)),p1) is non empty functional closed closed compact V258( TOP-REAL 2) V258( TOP-REAL 2) being_simple_closed_curve Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : |.(b₁ - (0. (TOP-REAL 2))).| = p1 } is set
C is V16() Function-like FinSequence-like Element of the carrier of (2)
(C) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
Cm is V16() Function-like FinSequence-like Element of the carrier of (2)
(Cm) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
{Cm} is non empty trivial functional compact Element of K6( the carrier of (2))
(Sphere ((0. (TOP-REAL 2)),p1)) \ {Cm} is functional Element of K6( the carrier of (TOP-REAL 2))
{C} is non empty trivial functional compact Element of K6( the carrier of (2))
(Sphere ((0. (TOP-REAL 2)),p1)) \ {C} is functional Element of K6( the carrier of (TOP-REAL 2))
the carrier of (2,(0. (TOP-REAL 2)),1) is non empty set
K6( the carrier of (2,(0. (TOP-REAL 2)),1)) is non empty set
E is Element of K6( the carrier of (2,(0. (TOP-REAL 2)),1))
(2,(0. (TOP-REAL 2)),1) | E is strict TopSpace-like V221() V222() SubSpace of (2,(0. (TOP-REAL 2)),1)
j is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
{j} is non empty trivial functional compact Element of K6( the carrier of (TOP-REAL 2))
(Sphere ((0. (TOP-REAL 2)),p1)) \ {j} is functional Element of K6( the carrier of (TOP-REAL 2))
(TOP-REAL 2) | ((Sphere ((0. (TOP-REAL 2)),p1)) \ {j}) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
Q is Element of K6( the carrier of (2,(0. (TOP-REAL 2)),1))
(2,(0. (TOP-REAL 2)),1) | Q is strict TopSpace-like V221() V222() SubSpace of (2,(0. (TOP-REAL 2)),1)
E is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
{E} is non empty trivial functional compact Element of K6( the carrier of (TOP-REAL 2))
(Sphere ((0. (TOP-REAL 2)),p1)) \ {E} is functional Element of K6( the carrier of (TOP-REAL 2))
(TOP-REAL 2) | ((Sphere ((0. (TOP-REAL 2)),p1)) \ {E}) is strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
K7( the carrier of R^1, the carrier of (2)) is non empty set
K6(K7( the carrier of R^1, the carrier of (2))) is non empty set
A1 is complex real ext-real Element of the carrier of R^1
(2 * PI) * A1 is complex real ext-real Element of REAL
cos ((2 * PI) * A1) is complex real ext-real Element of REAL
sin ((2 * PI) * A1) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| - |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
- |[0,0]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K543(|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K394((TOP-REAL 2),|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
the U6 of (TOP-REAL 2) is non empty V16() V19(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2))) V20( the carrier of (TOP-REAL 2)) Function-like total quasi_total Function-yielding V312() Element of K6(K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2)))
K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)) is non empty set
K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2)) is non empty set
K6(K7(K7( the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2)), the carrier of (TOP-REAL 2))) is non empty set
K414( the carrier of (TOP-REAL 2), the U6 of (TOP-REAL 2),|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]|,(- |[0,0]|)) is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K541(|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]|,(- |[0,0]|)) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
K545(|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]|,|[0,0]|) is V16() V19( NAT ) V20( REAL ) Function-like V51() V52() V53() FinSequence-like FinSequence of REAL
|.(|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| - |[0,0]|).| is complex real ext-real non negative Element of REAL
|.|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]|.| is complex real ext-real non negative Element of REAL
|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `1 is complex real ext-real Element of REAL
(|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `1) ^2 is complex real ext-real Element of REAL
(|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `1) * (|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `1) is complex real ext-real set
|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `2 is complex real ext-real Element of REAL
(|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `2) ^2 is complex real ext-real Element of REAL
(|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `2) * (|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `2) is complex real ext-real set
((|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `1) ^2) + ((|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `2) ^2) is complex real ext-real Element of REAL
sqrt (((|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `1) ^2) + ((|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `2) ^2)) is complex real ext-real Element of REAL
(cos ((2 * PI) * A1)) ^2 is complex real ext-real Element of REAL
(cos ((2 * PI) * A1)) * (cos ((2 * PI) * A1)) is complex real ext-real set
((cos ((2 * PI) * A1)) ^2) + ((|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `2) ^2) is complex real ext-real Element of REAL
sqrt (((cos ((2 * PI) * A1)) ^2) + ((|[(cos ((2 * PI) * A1)),(sin ((2 * PI) * A1))]| `2) ^2)) is complex real ext-real Element of REAL
(sin ((2 * PI) * A1)) ^2 is complex real ext-real Element of REAL
(sin ((2 * PI) * A1)) * (sin ((2 * PI) * A1)) is complex real ext-real set
((cos ((2 * PI) * A1)) ^2) + ((sin ((2 * PI) * A1)) ^2) is complex real ext-real Element of REAL
sqrt (((cos ((2 * PI) * A1)) ^2) + ((sin ((2 * PI) * A1)) ^2)) is complex real ext-real Element of REAL
A1 is non empty V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like total quasi_total Element of K6(K7( the carrier of R^1, the carrier of (2)))
C is complex real ext-real set
A1 . C is set
(2 * PI) * C is complex real ext-real Element of REAL
cos ((2 * PI) * C) is complex real ext-real Element of REAL
sin ((2 * PI) * C) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * C)),(sin ((2 * PI) * C))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A1 is non empty V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like total quasi_total Element of K6(K7( the carrier of R^1, the carrier of (2)))
C is non empty V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like total quasi_total Element of K6(K7( the carrier of R^1, the carrier of (2)))
Cm is complex real ext-real Element of the carrier of R^1
A1 . Cm is V16() Function-like FinSequence-like Element of the carrier of (2)
C . Cm is V16() Function-like FinSequence-like Element of the carrier of (2)
(2 * PI) * Cm is complex real ext-real Element of REAL
cos ((2 * PI) * Cm) is complex real ext-real Element of REAL
sin ((2 * PI) * Cm) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * Cm)),(sin ((2 * PI) * Cm))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
() is non empty V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like total quasi_total Element of K6(K7( the carrier of R^1, the carrier of (2)))
dom () is V67() V68() V69() Element of K6( the carrier of R^1)
A1 is complex real ext-real integer set
C is complex real ext-real set
() . C is set
C + A1 is complex real ext-real set
() . (C + A1) is set
Cm is complex real ext-real integer set
C + Cm is complex real ext-real set
() . (C + Cm) is set
Cm - 1 is complex real ext-real integer Element of REAL
C + (Cm - 1) is complex real ext-real set
() . (C + (Cm - 1)) is set
Cm + 1 is complex real ext-real integer Element of REAL
C + (Cm + 1) is complex real ext-real set
() . (C + (Cm + 1)) is set
C + (Cm - 1) is complex real ext-real Element of REAL
() . (C + (Cm - 1)) is set
(C + Cm) - 1 is complex real ext-real Element of REAL
(2 * PI) * ((C + Cm) - 1) is complex real ext-real Element of REAL
cos ((2 * PI) * ((C + Cm) - 1)) is complex real ext-real Element of REAL
sin ((2 * PI) * ((C + Cm) - 1)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * ((C + Cm) - 1))),(sin ((2 * PI) * ((C + Cm) - 1)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(2 * PI) * (C + Cm) is complex real ext-real Element of REAL
cos ((2 * PI) * (C + Cm)) is complex real ext-real Element of REAL
(2 * PI) * (- 1) is non empty complex real ext-real non positive negative Element of REAL
((2 * PI) * (C + Cm)) + ((2 * PI) * (- 1)) is complex real ext-real Element of REAL
sin (((2 * PI) * (C + Cm)) + ((2 * PI) * (- 1))) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (C + Cm))),(sin (((2 * PI) * (C + Cm)) + ((2 * PI) * (- 1))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
sin ((2 * PI) * (C + Cm)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (C + Cm))),(sin ((2 * PI) * (C + Cm)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
C + (Cm + 1) is complex real ext-real Element of REAL
() . (C + (Cm + 1)) is set
(C + Cm) + 1 is complex real ext-real Element of REAL
(2 * PI) * ((C + Cm) + 1) is complex real ext-real Element of REAL
cos ((2 * PI) * ((C + Cm) + 1)) is complex real ext-real Element of REAL
sin ((2 * PI) * ((C + Cm) + 1)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * ((C + Cm) + 1))),(sin ((2 * PI) * ((C + Cm) + 1)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
((2 * PI) * (C + Cm)) + ((2 * PI) * 1) is complex real ext-real Element of REAL
sin (((2 * PI) * (C + Cm)) + ((2 * PI) * 1)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (C + Cm))),(sin (((2 * PI) * (C + Cm)) + ((2 * PI) * 1)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
C + 0 is complex real ext-real set
() . (C + 0) is set
A1 is complex real ext-real integer set
() . A1 is set
(2 * PI) * A1 is complex real ext-real Element of REAL
((2 * PI) * A1) + 0 is complex real ext-real Element of REAL
cos (((2 * PI) * A1) + 0) is complex real ext-real Element of REAL
sin ((2 * PI) * A1) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * A1) + 0)),(sin ((2 * PI) * A1))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
sin (((2 * PI) * A1) + 0) is complex real ext-real Element of REAL
|[(cos 0),(sin (((2 * PI) * A1) + 0))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
{()} is non empty trivial functional compact Element of K6( the carrier of (2))
() " {()} is V67() V68() V69() Element of K6( the carrier of R^1)
A1 is set
() . A1 is set
C is complex real ext-real Element of REAL
(2 * PI) * C is complex real ext-real Element of REAL
cos ((2 * PI) * C) is complex real ext-real Element of REAL
sin ((2 * PI) * C) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * C)),(sin ((2 * PI) * C))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A1 is set
C is complex real ext-real integer set
() . C is set
() . (1 / 2) is set
(2 * PI) * (1 / 2) is non empty complex real ext-real positive non negative Element of REAL
cos ((2 * PI) * (1 / 2)) is complex real ext-real Element of REAL
sin ((2 * PI) * (1 / 2)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (1 / 2))),(sin ((2 * PI) * (1 / 2)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A1 is complex real ext-real set
frac A1 is complex real ext-real Element of REAL
() . A1 is set
[\A1/] is complex real ext-real integer set
- [\A1/] is complex real ext-real integer set
A1 + (- [\A1/]) is complex real ext-real set
() . (A1 + (- [\A1/])) is set
A1 - [\A1/] is complex real ext-real set
() . (A1 - [\A1/]) is set
4 is non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() Element of NAT
1 / 4 is non empty complex real ext-real positive non negative Element of REAL
|[0,1]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A1 is complex real ext-real set
frac A1 is complex real ext-real Element of REAL
() . A1 is set
[\A1/] is complex real ext-real integer set
A1 - [\A1/] is complex real ext-real set
- [\A1/] is complex real ext-real integer set
A1 + (- [\A1/]) is complex real ext-real set
() . (A1 + (- [\A1/])) is set
(2 * PI) * (1 / 4) is non empty complex real ext-real positive non negative Element of REAL
cos ((2 * PI) * (1 / 4)) is complex real ext-real Element of REAL
sin ((2 * PI) * (1 / 4)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (1 / 4))),(sin ((2 * PI) * (1 / 4)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
3 / 4 is non empty complex real ext-real positive non negative Element of REAL
|[0,(- 1)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A1 is complex real ext-real set
frac A1 is complex real ext-real Element of REAL
() . A1 is set
[\A1/] is complex real ext-real integer set
A1 - [\A1/] is complex real ext-real set
- [\A1/] is complex real ext-real integer set
A1 + (- [\A1/]) is complex real ext-real set
() . (A1 + (- [\A1/])) is set
(2 * PI) * (3 / 4) is non empty complex real ext-real positive non negative Element of REAL
cos ((2 * PI) * (3 / 4)) is complex real ext-real Element of REAL
sin ((2 * PI) * (3 / 4)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (3 / 4))),(sin ((2 * PI) * (3 / 4)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
AffineMap ((2 * PI),0) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
cos * (AffineMap ((2 * PI),0)) is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
sin * (AffineMap ((2 * PI),0)) is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
A1 is complex real ext-real set
() . A1 is set
(cos * (AffineMap ((2 * PI),0))) . A1 is complex real ext-real set
(sin * (AffineMap ((2 * PI),0))) . A1 is complex real ext-real set
|[((cos * (AffineMap ((2 * PI),0))) . A1),((sin * (AffineMap ((2 * PI),0))) . A1)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(2 * PI) * A1 is complex real ext-real Element of REAL
((2 * PI) * A1) + 0 is complex real ext-real Element of REAL
cos (((2 * PI) * A1) + 0) is complex real ext-real Element of REAL
sin ((2 * PI) * A1) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * A1) + 0)),(sin ((2 * PI) * A1))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
sin (((2 * PI) * A1) + 0) is complex real ext-real Element of REAL
|[((cos * (AffineMap ((2 * PI),0))) . A1),(sin (((2 * PI) * A1) + 0))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A1 is complex real ext-real set
() . A1 is set
C is complex real ext-real integer set
(2 * PI) * C is complex real ext-real Element of REAL
AffineMap ((2 * PI),((2 * PI) * C)) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
cos * (AffineMap ((2 * PI),((2 * PI) * C))) is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
(cos * (AffineMap ((2 * PI),((2 * PI) * C)))) . A1 is complex real ext-real set
Cm is complex real ext-real integer set
(2 * PI) * Cm is complex real ext-real Element of REAL
AffineMap ((2 * PI),((2 * PI) * Cm)) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
sin * (AffineMap ((2 * PI),((2 * PI) * Cm))) is non empty V16() V19( REAL ) V20( REAL ) Function-like total quasi_total continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
(sin * (AffineMap ((2 * PI),((2 * PI) * Cm)))) . A1 is complex real ext-real set
|[((cos * (AffineMap ((2 * PI),((2 * PI) * C)))) . A1),((sin * (AffineMap ((2 * PI),((2 * PI) * Cm)))) . A1)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(2 * PI) * A1 is complex real ext-real Element of REAL
((2 * PI) * A1) + 0 is complex real ext-real Element of REAL
cos (((2 * PI) * A1) + 0) is complex real ext-real Element of REAL
sin ((2 * PI) * A1) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * A1) + 0)),(sin ((2 * PI) * A1))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
((2 * PI) * A1) + ((2 * PI) * C) is complex real ext-real Element of REAL
cos (((2 * PI) * A1) + ((2 * PI) * C)) is complex real ext-real Element of REAL
sin (((2 * PI) * A1) + 0) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * A1) + ((2 * PI) * C))),(sin (((2 * PI) * A1) + 0))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
((2 * PI) * A1) + ((2 * PI) * Cm) is complex real ext-real Element of REAL
sin (((2 * PI) * A1) + ((2 * PI) * Cm)) is complex real ext-real Element of REAL
|[(cos (((2 * PI) * A1) + ((2 * PI) * C))),(sin (((2 * PI) * A1) + ((2 * PI) * Cm)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[((cos * (AffineMap ((2 * PI),((2 * PI) * C)))) . A1),(sin (((2 * PI) * A1) + ((2 * PI) * Cm)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(sin) is V16() V19( the carrier of (R^1 | ((dom sin)))) V20( the carrier of (R^1 | ((rng sin)))) Function-like quasi_total onto V51() V52() V53() continuous Element of K6(K7( the carrier of (R^1 | ((dom sin))), the carrier of (R^1 | ((rng sin)))))
((dom sin)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom sin)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom sin))) is V67() V68() V69() set
rng sin is V67() V68() V69() Element of K6(REAL)
((rng sin)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng sin)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng sin))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom sin))), the carrier of (R^1 | ((rng sin)))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom sin))), the carrier of (R^1 | ((rng sin))))) is non empty set
(cos) is V16() V19( the carrier of (R^1 | ((dom cos)))) V20( the carrier of (R^1 | ((rng cos)))) Function-like quasi_total onto V51() V52() V53() continuous Element of K6(K7( the carrier of (R^1 | ((dom cos))), the carrier of (R^1 | ((rng cos)))))
((dom cos)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom cos)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom cos))) is V67() V68() V69() set
rng cos is V67() V68() V69() Element of K6(REAL)
((rng cos)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng cos)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng cos))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom cos))), the carrier of (R^1 | ((rng cos)))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom cos))), the carrier of (R^1 | ((rng cos))))) is non empty set
dom (AffineMap ((2 * PI),0)) is V67() V68() V69() Element of K6(REAL)
((AffineMap ((2 * PI),0))) is V16() V19( the carrier of (R^1 | ((dom (AffineMap ((2 * PI),0)))))) V19( the carrier of (R^1 | ((dom (AffineMap ((2 * PI),0)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((2 * PI),0)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((2 * PI),0)))))) Function-like one-to-one quasi_total quasi_total onto onto bijective bijective V51() V52() V53() continuous continuous open Element of K6(K7( the carrier of (R^1 | ((dom (AffineMap ((2 * PI),0))))), the carrier of (R^1 | ((rng (AffineMap ((2 * PI),0)))))))
((dom (AffineMap ((2 * PI),0)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom (AffineMap ((2 * PI),0)))) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom (AffineMap ((2 * PI),0))))) is V67() V68() V69() set
rng (AffineMap ((2 * PI),0)) is V67() V68() V69() Element of K6(REAL)
((rng (AffineMap ((2 * PI),0)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng (AffineMap ((2 * PI),0)))) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng (AffineMap ((2 * PI),0))))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom (AffineMap ((2 * PI),0))))), the carrier of (R^1 | ((rng (AffineMap ((2 * PI),0)))))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom (AffineMap ((2 * PI),0))))), the carrier of (R^1 | ((rng (AffineMap ((2 * PI),0))))))) is non empty set
K7( the carrier of R^1, the carrier of (TOP-REAL 2)) is non empty set
K6(K7( the carrier of R^1, the carrier of (TOP-REAL 2))) is non empty set
A1 is non empty V16() V19( the carrier of R^1) V20( the carrier of R^1) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of R^1, the carrier of R^1))
Q is non empty V16() V19( the carrier of R^1) V20( the carrier of R^1) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of R^1, the carrier of R^1))
Q * A1 is non empty V16() V19( the carrier of R^1) V20( the carrier of R^1) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of R^1, the carrier of R^1))
A is non empty V16() V19( the carrier of R^1) V20( the carrier of R^1) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of R^1, the carrier of R^1))
A * A1 is non empty V16() V19( the carrier of R^1) V20( the carrier of R^1) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of R^1, the carrier of R^1))
E is non empty V16() V19( the carrier of R^1) V20( the carrier of (TOP-REAL 2)) Function-like total quasi_total Function-yielding V312() Element of K6(K7( the carrier of R^1, the carrier of (TOP-REAL 2)))
E is complex real ext-real Element of the carrier of R^1
E . E is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Af is functional Element of K6( the carrier of (TOP-REAL 2))
Int Af is functional open Element of K6( the carrier of (TOP-REAL 2))
Af is Element of the carrier of (Euclid 2)
Af is complex real ext-real set
Ball (Af,Af) is Element of K6( the carrier of (Euclid 2))
K6( the carrier of (Euclid 2)) is non empty set
(E . E) `2 is complex real ext-real Element of REAL
Af / (sqrt 2) is complex real ext-real Element of REAL
((E . E) `2) - (Af / (sqrt 2)) is complex real ext-real Element of REAL
((E . E) `2) + (Af / (sqrt 2)) is complex real ext-real Element of REAL
].(((E . E) `2) - (Af / (sqrt 2))),(((E . E) `2) + (Af / (sqrt 2))).[ is open V67() V68() V69() Element of K6(REAL)
(E . E) `1 is complex real ext-real Element of REAL
((E . E) `1) - (Af / (sqrt 2)) is complex real ext-real Element of REAL
((E . E) `1) + (Af / (sqrt 2)) is complex real ext-real Element of REAL
].(((E . E) `1) - (Af / (sqrt 2))),(((E . E) `1) + (Af / (sqrt 2))).[ is open V67() V68() V69() Element of K6(REAL)
(1,2) --> (].(((E . E) `1) - (Af / (sqrt 2))),(((E . E) `1) + (Af / (sqrt 2))).[,].(((E . E) `2) - (Af / (sqrt 2))),(((E . E) `2) + (Af / (sqrt 2))).[) is non empty V16() V19({1,2}) V20(K6(REAL)) Function-like total quasi_total Element of K6(K7({1,2},K6(REAL)))
{1,2} is non empty V67() V68() V69() V70() V71() V72() set
K7({1,2},K6(REAL)) is non empty set
K6(K7({1,2},K6(REAL))) is non empty set
c is V67() V68() V69() Element of K6( the carrier of R^1)
product ((1,2) --> (].(((E . E) `1) - (Af / (sqrt 2))),(((E . E) `1) + (Af / (sqrt 2))).[,].(((E . E) `2) - (Af / (sqrt 2))),(((E . E) `2) + (Af / (sqrt 2))).[)) is set
(Q * A1) . E is complex real ext-real Element of the carrier of R^1
(A * A1) . E is complex real ext-real Element of the carrier of R^1
|[((Q * A1) . E),((A * A1) . E)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
c is V67() V68() V69() Element of K6( the carrier of R^1)
(A * A1) .: c is V67() V68() V69() Element of K6( the carrier of R^1)
ac is V67() V68() V69() Element of K6( the carrier of R^1)
QQ is V67() V68() V69() Element of K6( the carrier of R^1)
(Q * A1) .: QQ is V67() V68() V69() Element of K6( the carrier of R^1)
c /\ QQ is V67() V68() V69() Element of K6( the carrier of R^1)
E .: (c /\ QQ) is functional Element of K6( the carrier of (TOP-REAL 2))
p is set
h1 is complex real ext-real Element of the carrier of R^1
E . h1 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(A * A1) . h1 is complex real ext-real Element of the carrier of R^1
(Q * A1) . h1 is complex real ext-real Element of the carrier of R^1
|[((Q * A1) . h1),((A * A1) . h1)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(1,2) --> (((Q * A1) . h1),((A * A1) . h1)) is non empty V16() V19({1,2}) V20( the carrier of R^1) Function-like total quasi_total V51() V52() V53() Element of K6(K7({1,2}, the carrier of R^1))
K7({1,2}, the carrier of R^1) is non empty V51() V52() V53() set
K6(K7({1,2}, the carrier of R^1)) is non empty set
A1 is V67() V68() V69() Element of K6( the carrier of R^1)
() | A1 is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
R^1 | A1 is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | A1) is V67() V68() V69() set
K7( the carrier of (R^1 | A1), the carrier of (2)) is set
K6(K7( the carrier of (R^1 | A1), the carrier of (2))) is non empty set
rng (() | A1) is Element of K6( the carrier of (2))
dom (() | A1) is V67() V68() V69() Element of K6( the carrier of R^1)
A1 is complex real ext-real set
arccos A1 is complex real ext-real Element of REAL
(arccos A1) / (2 * PI) is complex real ext-real Element of REAL
(1 * PI) / (2 * PI) is non empty complex real ext-real positive non negative Element of REAL
[.0,1.[ is non empty V67() V68() V69() Element of K6(REAL)
A1 is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | A1 is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | A1) is V67() V68() V69() set
K7( the carrier of (R^1 | A1), the carrier of (2)) is set
K6(K7( the carrier of (R^1 | A1), the carrier of (2))) is non empty set
() | A1 is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
C is V16() V19( the carrier of (R^1 | A1)) V20( the carrier of (2)) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | A1), the carrier of (2)))
dom C is V67() V68() V69() Element of K6( the carrier of (R^1 | A1))
K6( the carrier of (R^1 | A1)) is non empty set
rng C is Element of K6( the carrier of (2))
Cm is set
A is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
A `1 is complex real ext-real Element of REAL
A `2 is complex real ext-real Element of REAL
arccos (A `1) is complex real ext-real Element of REAL
(arccos (A `1)) / (2 * PI) is complex real ext-real Element of REAL
(A `1) ^2 is complex real ext-real Element of REAL
(A `1) * (A `1) is complex real ext-real set
(A `2) ^2 is complex real ext-real Element of REAL
(A `2) * (A `2) is complex real ext-real set
((A `1) ^2) + ((A `2) ^2) is complex real ext-real Element of REAL
|.A.| is complex real ext-real non negative Element of REAL
|.A.| ^2 is complex real ext-real Element of REAL
|.A.| * |.A.| is complex real ext-real non negative set
1 - ((arccos (A `1)) / (2 * PI)) is complex real ext-real Element of REAL
1 - 1 is complex real ext-real integer Element of REAL
C . (1 - ((arccos (A `1)) / (2 * PI))) is set
- ((arccos (A `1)) / (2 * PI)) is complex real ext-real Element of REAL
(- ((arccos (A `1)) / (2 * PI))) + 1 is complex real ext-real Element of REAL
() . ((- ((arccos (A `1)) / (2 * PI))) + 1) is set
() . (- ((arccos (A `1)) / (2 * PI))) is set
(2 * PI) * ((arccos (A `1)) / (2 * PI)) is complex real ext-real Element of REAL
- ((2 * PI) * ((arccos (A `1)) / (2 * PI))) is complex real ext-real Element of REAL
cos (- ((2 * PI) * ((arccos (A `1)) / (2 * PI)))) is complex real ext-real Element of REAL
(2 * PI) * (- ((arccos (A `1)) / (2 * PI))) is complex real ext-real Element of REAL
sin ((2 * PI) * (- ((arccos (A `1)) / (2 * PI)))) is complex real ext-real Element of REAL
|[(cos (- ((2 * PI) * ((arccos (A `1)) / (2 * PI))))),(sin ((2 * PI) * (- ((arccos (A `1)) / (2 * PI)))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
cos ((2 * PI) * ((arccos (A `1)) / (2 * PI))) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * ((arccos (A `1)) / (2 * PI)))),(sin ((2 * PI) * (- ((arccos (A `1)) / (2 * PI)))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
cos (arccos (A `1)) is complex real ext-real Element of REAL
sin (- ((2 * PI) * ((arccos (A `1)) / (2 * PI)))) is complex real ext-real Element of REAL
|[(cos (arccos (A `1))),(sin (- ((2 * PI) * ((arccos (A `1)) / (2 * PI)))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
sin ((2 * PI) * ((arccos (A `1)) / (2 * PI))) is complex real ext-real Element of REAL
- (sin ((2 * PI) * ((arccos (A `1)) / (2 * PI)))) is complex real ext-real Element of REAL
|[(cos (arccos (A `1))),(- (sin ((2 * PI) * ((arccos (A `1)) / (2 * PI)))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
sin (arccos (A `1)) is complex real ext-real Element of REAL
- (sin (arccos (A `1))) is complex real ext-real Element of REAL
|[(cos (arccos (A `1))),(- (sin (arccos (A `1))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[(A `1),(- (sin (arccos (A `1))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
- (A `2) is complex real ext-real Element of REAL
- (- (A `2)) is complex real ext-real Element of REAL
|[(A `1),(- (- (A `2)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
C . ((arccos (A `1)) / (2 * PI)) is set
() . ((arccos (A `1)) / (2 * PI)) is set
(2 * PI) * ((arccos (A `1)) / (2 * PI)) is complex real ext-real Element of REAL
cos ((2 * PI) * ((arccos (A `1)) / (2 * PI))) is complex real ext-real Element of REAL
sin ((2 * PI) * ((arccos (A `1)) / (2 * PI))) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * ((arccos (A `1)) / (2 * PI)))),(sin ((2 * PI) * ((arccos (A `1)) / (2 * PI))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
cos (arccos (A `1)) is complex real ext-real Element of REAL
|[(cos (arccos (A `1))),(sin ((2 * PI) * ((arccos (A `1)) / (2 * PI))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
sin (arccos (A `1)) is complex real ext-real Element of REAL
|[(cos (arccos (A `1))),(sin (arccos (A `1)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[(A `1),(sin (arccos (A `1)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[(A `1),(A `2)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
R^1 | ([#] R^1) is non empty strict TopSpace-like V299() SubSpace of R^1
() | REAL is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
() | [.0,1.[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
sin | [.(PI / 2),((3 / 2) * PI).] is V16() V19( REAL ) V20( REAL ) Function-like one-to-one continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
A1 is set
K121((() | [.0,1.[)) is set
C is set
(() | [.0,1.[) . A1 is set
(() | [.0,1.[) . C is set
[.0,(PI / 2).] is non empty closed V67() V68() V69() Element of K6(REAL)
dom (() | [.0,1.[) is V67() V68() V69() Element of K6( the carrier of R^1)
A is complex real ext-real Element of REAL
(() | [.0,1.[) . A is set
() . A is set
(2 * PI) * A is complex real ext-real Element of REAL
cos ((2 * PI) * A) is complex real ext-real Element of REAL
sin ((2 * PI) * A) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * A)),(sin ((2 * PI) * A))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Q is complex real ext-real Element of REAL
(() | [.0,1.[) . Q is set
() . Q is set
(2 * PI) * Q is complex real ext-real Element of REAL
cos ((2 * PI) * Q) is complex real ext-real Element of REAL
sin ((2 * PI) * Q) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * Q)),(sin ((2 * PI) * Q))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
cos . ((2 * PI) * Q) is complex real ext-real Element of REAL
cos . ((2 * PI) * A) is complex real ext-real Element of REAL
sin . ((2 * PI) * Q) is complex real ext-real Element of REAL
sin . ((2 * PI) * A) is complex real ext-real Element of REAL
(2 * PI) * (1 / 4) is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
((2 * PI) * 1) / 4 is non empty complex real ext-real positive non negative Element of REAL
[.0,(((2 * PI) * 1) / 4).] is non empty closed V67() V68() V69() Element of K6(REAL)
sin | [.0,(PI / 2).] is V16() V19( REAL ) V20( REAL ) Function-like one-to-one continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (sin | [.0,(PI / 2).]) is V67() V68() V69() Element of K6(REAL)
(sin | [.0,(PI / 2).]) . ((2 * PI) * A) is complex real ext-real set
(sin | [.0,(PI / 2).]) . ((2 * PI) * Q) is complex real ext-real set
((2 * PI) * Q) / (2 * PI) is complex real ext-real Element of REAL
(2 * PI) * (3 / 4) is non empty complex real ext-real positive non negative Element of REAL
[.((3 / 2) * PI),(2 * PI).[ is V67() V68() V69() Element of K6(REAL)
(2 * PI) * (3 / 4) is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * (1 / 2) is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * (1 / 4) is non empty complex real ext-real positive non negative Element of REAL
].(PI / 2),((2 * PI) * (1 / 2)).] is V67() V68() V69() Element of K6(REAL)
].(PI / 2),PI.] is V67() V68() V69() Element of K6(REAL)
[.0,((2 * PI) * (1 / 4)).] is non empty closed V67() V68() V69() Element of K6(REAL)
].((2 * PI) * (1 / 4)),((2 * PI) * (1 / 2)).] is V67() V68() V69() Element of K6(REAL)
sin | ].(PI / 2),PI.] is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (sin | ].(PI / 2),PI.]) is V67() V68() V69() Element of K6(REAL)
(sin | ].(PI / 2),PI.]) . ((2 * PI) * A) is complex real ext-real set
(sin | ].(PI / 2),PI.]) . ((2 * PI) * Q) is complex real ext-real set
((2 * PI) * Q) / (2 * PI) is complex real ext-real Element of REAL
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * (3 / 4) is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * (1 / 2) is non empty complex real ext-real positive non negative Element of REAL
].PI,((2 * PI) * (3 / 4)).] is V67() V68() V69() Element of K6(REAL)
].PI,((3 / 2) * PI).] is V67() V68() V69() Element of K6(REAL)
sin | ].PI,((3 / 2) * PI).] is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (sin | ].PI,((3 / 2) * PI).]) is V67() V68() V69() Element of K6(REAL)
(sin | ].PI,((3 / 2) * PI).]) . ((2 * PI) * A) is complex real ext-real set
(sin | ].PI,((3 / 2) * PI).]) . ((2 * PI) * Q) is complex real ext-real set
((2 * PI) * Q) / (2 * PI) is complex real ext-real Element of REAL
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * (3 / 4) is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * (1 / 2) is non empty complex real ext-real positive non negative Element of REAL
].PI,((3 / 2) * PI).] is V67() V68() V69() Element of K6(REAL)
(2 * PI) * (1 / 2) is non empty complex real ext-real positive non negative Element of REAL
sin | ].((3 / 2) * PI),(2 * PI).[ is V16() V19( REAL ) V20( REAL ) Function-like one-to-one continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (sin | ].((3 / 2) * PI),(2 * PI).[) is V67() V68() V69() Element of K6(REAL)
(sin | ].((3 / 2) * PI),(2 * PI).[) . ((2 * PI) * A) is complex real ext-real set
(sin | ].((3 / 2) * PI),(2 * PI).[) . ((2 * PI) * Q) is complex real ext-real set
((2 * PI) * Q) / (2 * PI) is complex real ext-real Element of REAL
A1 is complex real ext-real set
A1 + 1 is complex real ext-real Element of REAL
[.A1,(A1 + 1).[ is non empty V67() V68() V69() Element of K6(REAL)
() | [.A1,(A1 + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
C is set
K121((() | [.A1,(A1 + 1).[)) is set
Cm is set
(() | [.A1,(A1 + 1).[) . C is set
(() | [.A1,(A1 + 1).[) . Cm is set
dom (() | [.A1,(A1 + 1).[) is V67() V68() V69() Element of K6( the carrier of R^1)
dom (() | [.0,1.[) is V67() V68() V69() Element of K6( the carrier of R^1)
j is complex real ext-real Element of REAL
E is complex real ext-real Element of REAL
frac E is complex real ext-real Element of REAL
[\E/] is complex real ext-real integer set
E - [\E/] is complex real ext-real Element of REAL
frac j is complex real ext-real Element of REAL
[\j/] is complex real ext-real integer set
j - [\j/] is complex real ext-real Element of REAL
(() | [.A1,(A1 + 1).[) . j is set
() . j is set
- [\j/] is complex real ext-real integer set
j + (- [\j/]) is complex real ext-real Element of REAL
() . (j + (- [\j/])) is set
(() | [.0,1.[) . (frac j) is set
(() | [.A1,(A1 + 1).[) . E is set
() . E is set
- [\E/] is complex real ext-real integer set
E + (- [\E/]) is complex real ext-real Element of REAL
() . (E + (- [\E/])) is set
(() | [.0,1.[) . (frac E) is set
A1 is complex real ext-real set
A1 + 1 is complex real ext-real Element of REAL
].A1,(A1 + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
() | ].A1,(A1 + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
[.A1,(A1 + 1).[ is non empty V67() V68() V69() Element of K6(REAL)
() | [.A1,(A1 + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
A1 is complex real ext-real set
C is complex real ext-real set
A1 - C is complex real ext-real set
(C,A1) is non empty open Element of K6(K6( the carrier of R^1))
{ ].(C + b₁),(A1 + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
Cm is set
() | Cm is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
A is complex real ext-real integer V66() Element of INT
C + A is complex real ext-real set
A1 + A is complex real ext-real set
].(C + A),(A1 + A).[ is open V67() V68() V69() Element of K6(REAL)
(C + A) + 1 is complex real ext-real Element of REAL
[.(C + A),((C + A) + 1).[ is non empty V67() V68() V69() Element of K6(REAL)
() | [.(C + A),((C + A) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
(A1 - C) + (C + A) is complex real ext-real set
1 + (C + A) is complex real ext-real Element of REAL
A1 is complex real ext-real set
C is complex real ext-real set
(A1,C) is non empty open Element of K6(K6( the carrier of R^1))
{ ].(A1 + b₁),(C + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
union (A1,C) is V67() V68() V69() Element of K6( the carrier of R^1)
() .: (union (A1,C)) is Element of K6( the carrier of (2))
A is set
() .: A is Element of K6( the carrier of (2))
Q is complex real ext-real integer V66() Element of INT
A1 + Q is complex real ext-real set
C + Q is complex real ext-real set
].(A1 + Q),(C + Q).[ is open V67() V68() V69() Element of K6(REAL)
E is set
j is complex real ext-real Element of the carrier of R^1
() . j is V16() Function-like FinSequence-like Element of the carrier of (2)
Q is set
E is complex real ext-real integer V66() Element of INT
A1 + E is complex real ext-real set
C + E is complex real ext-real set
].(A1 + E),(C + E).[ is open V67() V68() V69() Element of K6(REAL)
(C + E) + Q is complex real ext-real set
j + Q is complex real ext-real set
((C + E) + Q) - E is complex real ext-real set
(j + Q) - E is complex real ext-real set
() . ((j + Q) - E) is set
Q - E is complex real ext-real integer set
j + (Q - E) is complex real ext-real set
() . (j + (Q - E)) is set
(A1 + E) + Q is complex real ext-real set
((A1 + E) + Q) - E is complex real ext-real set
A1 is complex real ext-real Element of the carrier of R^1
A1 + 1 is complex real ext-real Element of REAL
].A1,(A1 + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
() | ].A1,(A1 + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
(].A1,(A1 + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].A1,(A1 + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].A1,(A1 + 1).[)) is non empty V67() V68() V69() set
() . A1 is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . A1)) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
the carrier of ((() . A1)) is non empty set
K7( the carrier of (R^1 | (].A1,(A1 + 1).[)), the carrier of ((() . A1))) is non empty set
K6(K7( the carrier of (R^1 | (].A1,(A1 + 1).[)), the carrier of ((() . A1)))) is non empty set
[.A1,(A1 + 1).[ is non empty V67() V68() V69() Element of K6(REAL)
() | [.A1,(A1 + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
dom (() | ].A1,(A1 + 1).[) is V67() V68() V69() Element of K6( the carrier of R^1)
{(() . A1)} is non empty trivial functional compact Element of K6( the carrier of (2))
the carrier of (2) \ {(() . A1)} is Element of K6( the carrier of (2))
rng (() | ].A1,(A1 + 1).[) is Element of K6( the carrier of (2))
j is set
dom (() | [.A1,(A1 + 1).[) is V67() V68() V69() Element of K6( the carrier of R^1)
A1 + 0 is complex real ext-real Element of REAL
Q is set
(() | ].A1,(A1 + 1).[) . Q is set
(() | [.A1,(A1 + 1).[) . Q is set
() . Q is set
(() | [.A1,(A1 + 1).[) . A1 is set
A1 is complex real ext-real set
- A1 is complex real ext-real set
AffineMap (1,(- A1)) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
C is complex real ext-real set
C + A1 is complex real ext-real set
(C + A1) + 1 is complex real ext-real Element of REAL
].(C + A1),((C + A1) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[ is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
rng ((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[) is V67() V68() V69() Element of K6(REAL)
C + 1 is complex real ext-real Element of REAL
].C,(C + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
dom (AffineMap (1,(- A1))) is V67() V68() V69() Element of K6(REAL)
dom ((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[) is V67() V68() V69() Element of K6(REAL)
Q is set
E is set
((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[) . E is complex real ext-real set
j is complex real ext-real Element of REAL
j - A1 is complex real ext-real Element of REAL
(C + A1) - A1 is complex real ext-real set
((C + A1) + 1) - A1 is complex real ext-real Element of REAL
(AffineMap (1,(- A1))) . j is complex real ext-real Element of REAL
1 * j is complex real ext-real Element of REAL
(1 * j) + (- A1) is complex real ext-real Element of REAL
Q is set
E is complex real ext-real Element of REAL
(C + 1) + A1 is complex real ext-real Element of REAL
E + A1 is complex real ext-real Element of REAL
((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[) . (E + A1) is complex real ext-real set
(AffineMap (1,(- A1))) . (E + A1) is complex real ext-real Element of REAL
1 * (E + A1) is complex real ext-real Element of REAL
(1 * (E + A1)) + (- A1) is complex real ext-real Element of REAL
A1 is complex real ext-real integer set
- A1 is complex real ext-real integer set
AffineMap (1,(- A1)) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
C is complex real ext-real set
C + A1 is complex real ext-real set
((C + A1)) is complex real ext-real Element of the carrier of R^1
(((C + A1))) is non empty V16() V19( the carrier of (R^1 | (].((C + A1)),(((C + A1)) + 1).[))) V20( the carrier of ((() . ((C + A1))))) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | (].((C + A1)),(((C + A1)) + 1).[)), the carrier of ((() . ((C + A1))))))
((C + A1)) + 1 is complex real ext-real Element of REAL
].((C + A1)),(((C + A1)) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].((C + A1)),(((C + A1)) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].((C + A1)),(((C + A1)) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].((C + A1)),(((C + A1)) + 1).[)) is non empty V67() V68() V69() set
() . ((C + A1)) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . ((C + A1)))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
the carrier of ((() . ((C + A1)))) is non empty set
K7( the carrier of (R^1 | (].((C + A1)),(((C + A1)) + 1).[)), the carrier of ((() . ((C + A1))))) is non empty set
K6(K7( the carrier of (R^1 | (].((C + A1)),(((C + A1)) + 1).[)), the carrier of ((() . ((C + A1)))))) is non empty set
() | ].((C + A1)),(((C + A1)) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
(C) is complex real ext-real Element of the carrier of R^1
(C) + 1 is complex real ext-real Element of REAL
].(C),((C) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].(C),((C) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(C),((C) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].(C),((C) + 1).[)) is non empty V67() V68() V69() set
() . (C) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . (C))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
the carrier of ((() . (C))) is non empty set
(C + A1) + 1 is complex real ext-real Element of REAL
].(C + A1),((C + A1) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[ is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
((C)) is non empty V16() V19( the carrier of (R^1 | (].(C),((C) + 1).[))) V20( the carrier of ((() . (C)))) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | (].(C),((C) + 1).[)), the carrier of ((() . (C)))))
K7( the carrier of (R^1 | (].(C),((C) + 1).[)), the carrier of ((() . (C)))) is non empty set
K6(K7( the carrier of (R^1 | (].(C),((C) + 1).[)), the carrier of ((() . (C))))) is non empty set
() | ].(C),((C) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
((C)) * ((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[) is V16() V19( REAL ) V20( the carrier of ((() . (C)))) Function-like Element of K6(K7(REAL, the carrier of ((() . (C)))))
K7(REAL, the carrier of ((() . (C)))) is non empty set
K6(K7(REAL, the carrier of ((() . (C))))) is non empty set
C + 1 is complex real ext-real Element of REAL
].C,(C + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
dom (((C + A1))) is V67() V68() V69() Element of K6( the carrier of (R^1 | (].((C + A1)),(((C + A1)) + 1).[)))
K6( the carrier of (R^1 | (].((C + A1)),(((C + A1)) + 1).[))) is non empty set
dom (AffineMap (1,(- A1))) is V67() V68() V69() Element of K6(REAL)
dom ((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[) is V67() V68() V69() Element of K6(REAL)
E is set
(((C + A1))) . E is set
(((C)) * ((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[)) . E is set
Af is complex real ext-real Element of REAL
((C + A1) + 1) - A1 is complex real ext-real Element of REAL
Af - A1 is complex real ext-real Element of REAL
(C + A1) - A1 is complex real ext-real set
((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[) . E is complex real ext-real set
((C)) . (((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[) . E) is set
(AffineMap (1,(- A1))) . E is complex real ext-real set
((C)) . ((AffineMap (1,(- A1))) . E) is set
1 * Af is complex real ext-real Element of REAL
(1 * Af) + (- A1) is complex real ext-real Element of REAL
((C)) . ((1 * Af) + (- A1)) is set
Af + (- A1) is complex real ext-real Element of REAL
() . (Af + (- A1)) is set
() . Af is set
rng ((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[) is V67() V68() V69() Element of K6(REAL)
dom ((C)) is V67() V68() V69() Element of K6( the carrier of (R^1 | (].(C),((C) + 1).[)))
K6( the carrier of (R^1 | (].(C),((C) + 1).[))) is non empty set
dom (((C)) * ((AffineMap (1,(- A1))) | ].(C + A1),((C + A1) + 1).[)) is V67() V68() V69() Element of K6(REAL)
A1 is complex real ext-real Element of the carrier of R^1
(A1) is non empty V16() V19( the carrier of (R^1 | (].A1,(A1 + 1).[))) V20( the carrier of ((() . A1))) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | (].A1,(A1 + 1).[)), the carrier of ((() . A1))))
A1 + 1 is complex real ext-real Element of REAL
].A1,(A1 + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].A1,(A1 + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].A1,(A1 + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].A1,(A1 + 1).[)) is non empty V67() V68() V69() set
() . A1 is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . A1)) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
the carrier of ((() . A1)) is non empty set
K7( the carrier of (R^1 | (].A1,(A1 + 1).[)), the carrier of ((() . A1))) is non empty set
K6(K7( the carrier of (R^1 | (].A1,(A1 + 1).[)), the carrier of ((() . A1)))) is non empty set
() | ].A1,(A1 + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
rng (A1) is Element of K6( the carrier of ((() . A1)))
K6( the carrier of ((() . A1))) is non empty set
E is set
[\A1/] is complex real ext-real integer set
dom (() | ].A1,(A1 + 1).[) is V67() V68() V69() Element of K6( the carrier of R^1)
j is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
j `1 is complex real ext-real Element of REAL
j `2 is complex real ext-real Element of REAL
arccos (j `1) is complex real ext-real Element of REAL
(arccos (j `1)) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos (j `1)) / (2 * PI)) is complex real ext-real Element of REAL
((arccos (j `1)) / (2 * PI)) - ((arccos (j `1)) / (2 * PI)) is complex real ext-real Element of REAL
(j `1) ^2 is complex real ext-real Element of REAL
(j `1) * (j `1) is complex real ext-real set
(j `2) ^2 is complex real ext-real Element of REAL
(j `2) * (j `2) is complex real ext-real set
((j `1) ^2) + ((j `2) ^2) is complex real ext-real Element of REAL
|.j.| is complex real ext-real non negative Element of REAL
|.j.| ^2 is complex real ext-real Element of REAL
|.j.| * |.j.| is complex real ext-real non negative set
- ((arccos (j `1)) / (2 * PI)) is complex real ext-real Element of REAL
() . (- ((arccos (j `1)) / (2 * PI))) is set
(2 * PI) * ((arccos (j `1)) / (2 * PI)) is complex real ext-real Element of REAL
- ((2 * PI) * ((arccos (j `1)) / (2 * PI))) is complex real ext-real Element of REAL
cos (- ((2 * PI) * ((arccos (j `1)) / (2 * PI)))) is complex real ext-real Element of REAL
(2 * PI) * (- ((arccos (j `1)) / (2 * PI))) is complex real ext-real Element of REAL
sin ((2 * PI) * (- ((arccos (j `1)) / (2 * PI)))) is complex real ext-real Element of REAL
|[(cos (- ((2 * PI) * ((arccos (j `1)) / (2 * PI))))),(sin ((2 * PI) * (- ((arccos (j `1)) / (2 * PI)))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
cos ((2 * PI) * ((arccos (j `1)) / (2 * PI))) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * ((arccos (j `1)) / (2 * PI)))),(sin ((2 * PI) * (- ((arccos (j `1)) / (2 * PI)))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
cos (arccos (j `1)) is complex real ext-real Element of REAL
sin (- ((2 * PI) * ((arccos (j `1)) / (2 * PI)))) is complex real ext-real Element of REAL
|[(cos (arccos (j `1))),(sin (- ((2 * PI) * ((arccos (j `1)) / (2 * PI)))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
sin ((2 * PI) * ((arccos (j `1)) / (2 * PI))) is complex real ext-real Element of REAL
- (sin ((2 * PI) * ((arccos (j `1)) / (2 * PI)))) is complex real ext-real Element of REAL
|[(cos (arccos (j `1))),(- (sin ((2 * PI) * ((arccos (j `1)) / (2 * PI)))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
sin (arccos (j `1)) is complex real ext-real Element of REAL
- (sin (arccos (j `1))) is complex real ext-real Element of REAL
|[(cos (arccos (j `1))),(- (sin (arccos (j `1))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[(j `1),(- (sin (arccos (j `1))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
- (j `2) is complex real ext-real Element of REAL
- (- (j `2)) is complex real ext-real Element of REAL
|[(j `1),(- (- (j `2)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(1 - ((arccos (j `1)) / (2 * PI))) + [\A1/] is complex real ext-real Element of REAL
(() | ].A1,(A1 + 1).[) . ((1 - ((arccos (j `1)) / (2 * PI))) + [\A1/]) is set
[\A1/] + 1 is complex real ext-real integer Element of REAL
(- ((arccos (j `1)) / (2 * PI))) + ([\A1/] + 1) is complex real ext-real Element of REAL
() . ((- ((arccos (j `1)) / (2 * PI))) + ([\A1/] + 1)) is set
(1 - ((arccos (j `1)) / (2 * PI))) + [\A1/] is complex real ext-real Element of REAL
A1 - 0 is complex real ext-real Element of REAL
[\A1/] - ((arccos (j `1)) / (2 * PI)) is complex real ext-real Element of REAL
(- ((arccos (j `1)) / (2 * PI))) + [\A1/] is complex real ext-real Element of REAL
((- ((arccos (j `1)) / (2 * PI))) + [\A1/]) + 1 is complex real ext-real Element of REAL
[\A1/] + 1 is complex real ext-real integer Element of REAL
([\A1/] + 1) + (1 - ((arccos (j `1)) / (2 * PI))) is complex real ext-real Element of REAL
([\A1/] + 1) + 0 is complex real ext-real integer Element of REAL
2 - ((arccos (j `1)) / (2 * PI)) is complex real ext-real Element of REAL
(2 - ((arccos (j `1)) / (2 * PI))) + [\A1/] is complex real ext-real Element of REAL
(- ((arccos (j `1)) / (2 * PI))) + ([\A1/] + 1) is complex real ext-real Element of REAL
() . ((- ((arccos (j `1)) / (2 * PI))) + ([\A1/] + 1)) is set
((1 - ((arccos (j `1)) / (2 * PI))) + [\A1/]) + 1 is complex real ext-real Element of REAL
((- ((arccos (j `1)) / (2 * PI))) + [\A1/]) + 2 is complex real ext-real Element of REAL
[\A1/] + 2 is complex real ext-real integer Element of REAL
(- ((arccos (j `1)) / (2 * PI))) + ([\A1/] + 2) is complex real ext-real Element of REAL
(() | ].A1,(A1 + 1).[) . ((- ((arccos (j `1)) / (2 * PI))) + ([\A1/] + 2)) is set
() . ((- ((arccos (j `1)) / (2 * PI))) + ([\A1/] + 2)) is set
(1 - ((arccos (j `1)) / (2 * PI))) + [\A1/] is complex real ext-real Element of REAL
() . ((arccos (j `1)) / (2 * PI)) is set
(2 * PI) * ((arccos (j `1)) / (2 * PI)) is complex real ext-real Element of REAL
cos ((2 * PI) * ((arccos (j `1)) / (2 * PI))) is complex real ext-real Element of REAL
sin ((2 * PI) * ((arccos (j `1)) / (2 * PI))) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * ((arccos (j `1)) / (2 * PI)))),(sin ((2 * PI) * ((arccos (j `1)) / (2 * PI))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
cos (arccos (j `1)) is complex real ext-real Element of REAL
|[(cos (arccos (j `1))),(sin ((2 * PI) * ((arccos (j `1)) / (2 * PI))))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
sin (arccos (j `1)) is complex real ext-real Element of REAL
|[(cos (arccos (j `1))),(sin (arccos (j `1)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[(j `1),(sin (arccos (j `1)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
|[(j `1),(j `2)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
((arccos (j `1)) / (2 * PI)) + [\A1/] is complex real ext-real Element of REAL
(() | ].A1,(A1 + 1).[) . (((arccos (j `1)) / (2 * PI)) + [\A1/]) is set
() . (((arccos (j `1)) / (2 * PI)) + [\A1/]) is set
((arccos (j `1)) / (2 * PI)) + [\A1/] is complex real ext-real Element of REAL
[\A1/] + 1 is complex real ext-real integer Element of REAL
0 + ([\A1/] + 1) is complex real ext-real integer Element of REAL
((arccos (j `1)) / (2 * PI)) + ([\A1/] + 1) is complex real ext-real Element of REAL
(((arccos (j `1)) / (2 * PI)) + [\A1/]) + 1 is complex real ext-real Element of REAL
1 + A1 is complex real ext-real Element of REAL
(() | ].A1,(A1 + 1).[) . ((((arccos (j `1)) / (2 * PI)) + [\A1/]) + 1) is set
() . (((arccos (j `1)) / (2 * PI)) + ([\A1/] + 1)) is set
((arccos (j `1)) / (2 * PI)) + [\A1/] is complex real ext-real Element of REAL
{(() . A1)} is non empty trivial functional compact Element of K6( the carrier of (2))
([#] (2)) \ {(() . A1)} is Element of K6( the carrier of (2))
(2) | (([#] (2)) \ {(() . A1)}) is strict TopSpace-like V221() V222() SubSpace of (2)
R^1 | (].0,1.[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].0,1.[)) is non empty V67() V68() V69() set
K7( the carrier of (()), the carrier of (R^1 | (].0,1.[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (()), the carrier of (R^1 | (].0,1.[)))) is non empty set
A1 is non empty V67() V68() V69() Element of K6( the carrier of R^1)
C is V16() Function-like FinSequence-like Element of the carrier of (())
the carrier of (2) \ {()} is Element of K6( the carrier of (2))
Cm is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
<*Cm,A*> is set
Q is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Q `2 is complex real ext-real Element of REAL
arccos Cm is complex real ext-real Element of REAL
Q `1 is complex real ext-real Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
|.Q.| is complex real ext-real non negative Element of REAL
|.Q.| ^2 is complex real ext-real Element of REAL
|.Q.| * |.Q.| is complex real ext-real non negative set
Cm ^2 is complex real ext-real Element of REAL
Cm * Cm is complex real ext-real set
A ^2 is complex real ext-real Element of REAL
A * A is complex real ext-real set
(Cm ^2) + (A ^2) is complex real ext-real Element of REAL
(arccos Cm) / (2 * PI) is complex real ext-real Element of REAL
1 - 1 is complex real ext-real integer Element of REAL
1 + (A ^2) is complex real ext-real Element of REAL
(1 + (A ^2)) - 1 is complex real ext-real Element of REAL
Q is complex real ext-real Element of the carrier of (R^1 | (].0,(0 + p1).[))
1 - ((arccos Cm) / (2 * PI)) is complex real ext-real Element of REAL
|[Cm,A]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(1 * PI) / (2 * PI) is non empty complex real ext-real positive non negative Element of REAL
1 - (1 / 2) is complex real ext-real Element of REAL
1 - ((1 * PI) / (2 * PI)) is complex real ext-real Element of REAL
Q is complex real ext-real Element of the carrier of (R^1 | (].0,(0 + p1).[))
1 - ((arccos Cm) / (2 * PI)) is complex real ext-real Element of REAL
|[Cm,A]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
1 - ((arccos Cm) / (2 * PI)) is complex real ext-real Element of REAL
(2 * PI) - (arccos Cm) is complex real ext-real Element of REAL
((2 * PI) - (arccos Cm)) / (2 * PI) is complex real ext-real Element of REAL
(2 * PI) / (2 * PI) is non empty complex real ext-real positive non negative Element of REAL
((2 * PI) / (2 * PI)) - ((arccos Cm) / (2 * PI)) is complex real ext-real Element of REAL
(2 * PI) - 0 is non empty complex real ext-real positive non negative Element of REAL
1 * (arccos Cm) is complex real ext-real Element of REAL
(arccos Cm) - (arccos Cm) is complex real ext-real Element of REAL
Q is complex real ext-real Element of the carrier of (R^1 | (].0,(0 + p1).[))
|[Cm,A]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K7( the carrier of (()), the carrier of (R^1 | (].0,(0 + p1).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (()), the carrier of (R^1 | (].0,(0 + p1).[)))) is non empty set
C is non empty V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].0,(0 + p1).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].0,(0 + p1).[))))
A1 is non empty V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].0,1.[))))
C is non empty V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].0,1.[))))
Cm is V16() Function-like FinSequence-like Element of the carrier of (())
C . Cm is complex real ext-real Element of the carrier of (R^1 | (].0,1.[))
A1 . Cm is complex real ext-real Element of the carrier of (R^1 | (].0,1.[))
A is complex real ext-real set
Q is complex real ext-real set
|[A,Q]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos A is complex real ext-real Element of REAL
(arccos A) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos A) / (2 * PI)) is complex real ext-real Element of REAL
E is complex real ext-real set
j is complex real ext-real set
|[E,j]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos E is complex real ext-real Element of REAL
(arccos E) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos E) / (2 * PI)) is complex real ext-real Element of REAL
() is non empty V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].0,1.[))))
(].(1 / 2),(3 / 2).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(1 / 2),(3 / 2).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].(1 / 2),(3 / 2).[)) is non empty V67() V68() V69() set
K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[)))) is non empty set
C is non empty V67() V68() V69() Element of K6( the carrier of R^1)
Cm is V16() Function-like FinSequence-like Element of the carrier of (())
{()} is non empty trivial functional compact Element of K6( the carrier of (2))
the carrier of (2) \ {()} is Element of K6( the carrier of (2))
A is complex real ext-real Element of REAL
Q is complex real ext-real Element of REAL
<*A,Q*> is set
E is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
E `2 is complex real ext-real Element of REAL
arccos A is complex real ext-real Element of REAL
E `1 is complex real ext-real Element of REAL
(arccos A) / (2 * PI) is complex real ext-real Element of REAL
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty ordinal natural complex real ext-real positive non negative integer set
|.E.| is complex real ext-real non negative Element of REAL
|.E.| ^2 is complex real ext-real Element of REAL
|.E.| * |.E.| is complex real ext-real non negative set
A ^2 is complex real ext-real Element of REAL
A * A is complex real ext-real set
Q ^2 is complex real ext-real Element of REAL
Q * Q is complex real ext-real set
(A ^2) + (Q ^2) is complex real ext-real Element of REAL
1 - 1 is complex real ext-real integer Element of REAL
(- 1) ^2 is complex real ext-real Element of REAL
(- 1) * (- 1) is non empty complex real ext-real positive non negative integer set
((- 1) ^2) + (Q ^2) is complex real ext-real Element of REAL
(((- 1) ^2) + (Q ^2)) - 1 is complex real ext-real Element of REAL
(1 ^2) + (Q ^2) is complex real ext-real Element of REAL
((1 ^2) + (Q ^2)) - 1 is complex real ext-real Element of REAL
((arccos A) / (2 * PI)) * 2 is complex real ext-real Element of REAL
(1 / 2) * 2 is non empty complex real ext-real positive non negative Element of REAL
(arccos A) / PI is complex real ext-real Element of REAL
Q is complex real ext-real set
1 + ((arccos A) / (2 * PI)) is complex real ext-real Element of REAL
1 + (1 / 2) is non empty complex real ext-real positive non negative Element of REAL
1 + 0 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
Q is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))
1 - ((arccos A) / (2 * PI)) is complex real ext-real Element of REAL
|[A,Q]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Q is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))
1 - ((arccos A) / (2 * PI)) is complex real ext-real Element of REAL
|[A,Q]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
1 - ((arccos A) / (2 * PI)) is complex real ext-real Element of REAL
1 - (1 / 2) is complex real ext-real Element of REAL
E is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))
|[A,Q]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)))) is non empty set
Cm is non empty V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))))
C is non empty V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
Cm is non empty V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
A is V16() Function-like FinSequence-like Element of the carrier of (())
Cm . A is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),(3 / 2).[))
C . A is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),(3 / 2).[))
Q is complex real ext-real set
E is complex real ext-real set
|[Q,E]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos Q is complex real ext-real Element of REAL
(arccos Q) / (2 * PI) is complex real ext-real Element of REAL
1 + ((arccos Q) / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos Q) / (2 * PI)) is complex real ext-real Element of REAL
j is complex real ext-real set
Q is complex real ext-real set
|[j,Q]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos j is complex real ext-real Element of REAL
(arccos j) / (2 * PI) is complex real ext-real Element of REAL
1 + ((arccos j) / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos j) / (2 * PI)) is complex real ext-real Element of REAL
() is non empty V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
(0) is complex real ext-real Element of the carrier of R^1
() . (0) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . (0))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
the carrier of ((() . (0))) is non empty set
(0) + 1 is complex real ext-real Element of REAL
].(0),((0) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].(0),((0) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(0),((0) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].(0),((0) + 1).[)) is non empty V67() V68() V69() set
((0)) is non empty V16() V19( the carrier of (R^1 | (].(0),((0) + 1).[))) V20( the carrier of ((() . (0)))) Function-like one-to-one total quasi_total onto bijective continuous Element of K6(K7( the carrier of (R^1 | (].(0),((0) + 1).[)), the carrier of ((() . (0)))))
K7( the carrier of (R^1 | (].(0),((0) + 1).[)), the carrier of ((() . (0)))) is non empty set
K6(K7( the carrier of (R^1 | (].(0),((0) + 1).[)), the carrier of ((() . (0))))) is non empty set
() | ].(0),((0) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
((0)) /" is non empty V16() V19( the carrier of ((() . (0)))) V20( the carrier of (R^1 | (].(0),((0) + 1).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((() . (0))), the carrier of (R^1 | (].(0),((0) + 1).[))))
K7( the carrier of ((() . (0))), the carrier of (R^1 | (].(0),((0) + 1).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((() . (0))), the carrier of (R^1 | (].(0),((0) + 1).[)))) is non empty set
A is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | A is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | A) is non empty V67() V68() V69() set
K7( the carrier of (R^1 | A), the carrier of (())) is non empty set
K6(K7( the carrier of (R^1 | A), the carrier of (()))) is non empty set
id the carrier of (R^1 | A) is non empty V16() V19( the carrier of (R^1 | A)) V20( the carrier of (R^1 | A)) Function-like one-to-one total quasi_total V51() V52() V53() V55() V57() Element of K6(K7( the carrier of (R^1 | A), the carrier of (R^1 | A)))
K7( the carrier of (R^1 | A), the carrier of (R^1 | A)) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | A), the carrier of (R^1 | A))) is non empty set
dom (id the carrier of (R^1 | A)) is V67() V68() V69() Element of K6( the carrier of (R^1 | A))
K6( the carrier of (R^1 | A)) is non empty set
Q is complex real ext-real Element of the carrier of R^1
() . Q is V16() Function-like FinSequence-like Element of the carrier of (2)
j is non empty V16() V19( the carrier of (R^1 | A)) V20( the carrier of (())) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | A), the carrier of (())))
rng j is Element of K6( the carrier of (()))
Af is set
() * j is non empty V16() V19( the carrier of (R^1 | A)) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | A), the carrier of (R^1 | (].0,1.[))))
K7( the carrier of (R^1 | A), the carrier of (R^1 | (].0,1.[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | A), the carrier of (R^1 | (].0,1.[)))) is non empty set
dom (() * j) is V67() V68() V69() Element of K6( the carrier of (R^1 | A))
Af is complex real ext-real Element of the carrier of (R^1 | A)
j . Af is V16() Function-like FinSequence-like Element of the carrier of (())
() . (j . Af) is complex real ext-real Element of the carrier of (R^1 | (].0,1.[))
L is complex real ext-real set
L is complex real ext-real set
|[L,L]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos L is complex real ext-real Element of REAL
(arccos L) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos L) / (2 * PI)) is complex real ext-real Element of REAL
() . Af is set
Af is complex real ext-real Element of REAL
(cos * (AffineMap ((2 * PI),0))) . Af is complex real ext-real Element of REAL
(sin * (AffineMap ((2 * PI),0))) . Af is complex real ext-real Element of REAL
|[((cos * (AffineMap ((2 * PI),0))) . Af),((sin * (AffineMap ((2 * PI),0))) . Af)]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(AffineMap ((2 * PI),0)) . Af is complex real ext-real Element of REAL
sin . ((AffineMap ((2 * PI),0)) . Af) is complex real ext-real Element of REAL
cos . ((AffineMap ((2 * PI),0)) . Af) is complex real ext-real Element of REAL
cos ((AffineMap ((2 * PI),0)) . Af) is complex real ext-real Element of REAL
(2 * PI) * Af is complex real ext-real Element of REAL
((2 * PI) * Af) + 0 is complex real ext-real Element of REAL
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
((AffineMap ((2 * PI),0)) . Af) / ((2 * PI) * 1) is complex real ext-real Element of REAL
Af / 1 is complex real ext-real Element of REAL
1 - Af is complex real ext-real Element of REAL
(AffineMap ((2 * PI),0)) . (1 - Af) is complex real ext-real Element of REAL
(2 * PI) * (1 - Af) is complex real ext-real Element of REAL
((2 * PI) * (1 - Af)) + 0 is complex real ext-real Element of REAL
((AffineMap ((2 * PI),0)) . (1 - Af)) / ((2 * PI) * 1) is complex real ext-real Element of REAL
(1 - Af) / 1 is complex real ext-real Element of REAL
((AffineMap ((2 * PI),0)) . Af) / (2 * PI) is complex real ext-real Element of REAL
(1 * PI) / (2 * PI) is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) / (2 * PI) is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * (1 / 2) is non empty complex real ext-real positive non negative Element of REAL
0 / (2 * PI) is empty trivial complex real ext-real non positive non negative Function-like functional V67() V68() V69() V70() V71() V72() V73() Element of REAL
((AffineMap ((2 * PI),0)) . Af) / (2 * PI) is complex real ext-real Element of REAL
(1 * PI) / (2 * PI) is non empty complex real ext-real positive non negative Element of REAL
1 - (1 / 2) is complex real ext-real Element of REAL
(2 * PI) * (1 / 2) is non empty complex real ext-real positive non negative Element of REAL
1 - 1 is complex real ext-real integer Element of REAL
cos . ((AffineMap ((2 * PI),0)) . (1 - Af)) is complex real ext-real Element of REAL
- ((2 * PI) * Af) is complex real ext-real Element of REAL
(- ((2 * PI) * Af)) + ((2 * PI) * 1) is complex real ext-real Element of REAL
cos ((- ((2 * PI) * Af)) + ((2 * PI) * 1)) is complex real ext-real Element of REAL
cos (- ((2 * PI) * Af)) is complex real ext-real Element of REAL
cos ((2 * PI) * Af) is complex real ext-real Element of REAL
cos ((AffineMap ((2 * PI),0)) . (1 - Af)) is complex real ext-real Element of REAL
arccos (cos ((AffineMap ((2 * PI),0)) . (1 - Af))) is complex real ext-real Element of REAL
(() * j) . Af is complex real ext-real set
(id the carrier of (R^1 | A)) . Af is complex real ext-real set
j " is V16() Function-like set
((1 / 2)) is complex real ext-real Element of the carrier of R^1
() . ((1 / 2)) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . ((1 / 2)))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
the carrier of ((() . ((1 / 2)))) is non empty set
((1 / 2)) + 1 is complex real ext-real Element of REAL
].((1 / 2)),(((1 / 2)) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].((1 / 2)),(((1 / 2)) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].((1 / 2)),(((1 / 2)) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].((1 / 2)),(((1 / 2)) + 1).[)) is non empty V67() V68() V69() set
(((1 / 2))) is non empty V16() V19( the carrier of (R^1 | (].((1 / 2)),(((1 / 2)) + 1).[))) V20( the carrier of ((() . ((1 / 2))))) Function-like one-to-one total quasi_total onto bijective continuous Element of K6(K7( the carrier of (R^1 | (].((1 / 2)),(((1 / 2)) + 1).[)), the carrier of ((() . ((1 / 2))))))
K7( the carrier of (R^1 | (].((1 / 2)),(((1 / 2)) + 1).[)), the carrier of ((() . ((1 / 2))))) is non empty set
K6(K7( the carrier of (R^1 | (].((1 / 2)),(((1 / 2)) + 1).[)), the carrier of ((() . ((1 / 2)))))) is non empty set
() | ].((1 / 2)),(((1 / 2)) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
(((1 / 2))) /" is non empty V16() V19( the carrier of ((() . ((1 / 2))))) V20( the carrier of (R^1 | (].((1 / 2)),(((1 / 2)) + 1).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((() . ((1 / 2)))), the carrier of (R^1 | (].((1 / 2)),(((1 / 2)) + 1).[))))
K7( the carrier of ((() . ((1 / 2)))), the carrier of (R^1 | (].((1 / 2)),(((1 / 2)) + 1).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((() . ((1 / 2)))), the carrier of (R^1 | (].((1 / 2)),(((1 / 2)) + 1).[)))) is non empty set
A is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | A is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | A) is non empty V67() V68() V69() set
K7( the carrier of (R^1 | A), the carrier of (())) is non empty set
K6(K7( the carrier of (R^1 | A), the carrier of (()))) is non empty set
id the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)) is non empty V16() V19( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) V20( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) Function-like one-to-one total quasi_total V51() V52() V53() V55() V57() Element of K6(K7( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))))
K7( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)))) is non empty set
dom (id the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) is V67() V68() V69() Element of K6( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)))
K6( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) is non empty set
j is non empty V16() V19( the carrier of (R^1 | A)) V20( the carrier of (())) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | A), the carrier of (())))
rng j is Element of K6( the carrier of (()))
E is set
() * j is non empty V16() V19( the carrier of (R^1 | A)) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | A), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
K7( the carrier of (R^1 | A), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | A), the carrier of (R^1 | (].(1 / 2),(3 / 2).[)))) is non empty set
dom (() * j) is V67() V68() V69() Element of K6( the carrier of (R^1 | A))
K6( the carrier of (R^1 | A)) is non empty set
Af is complex real ext-real Element of the carrier of (R^1 | A)
j . Af is V16() Function-like FinSequence-like Element of the carrier of (())
() . (j . Af) is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),(3 / 2).[))
Af is complex real ext-real set
L is complex real ext-real set
|[Af,L]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos Af is complex real ext-real Element of REAL
(arccos Af) / (2 * PI) is complex real ext-real Element of REAL
1 + ((arccos Af) / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos Af) / (2 * PI)) is complex real ext-real Element of REAL
() . Af is set
Af is complex real ext-real Element of REAL
(2 * PI) * Af is complex real ext-real Element of REAL
cos ((2 * PI) * Af) is complex real ext-real Element of REAL
sin ((2 * PI) * Af) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * Af)),(sin ((2 * PI) * Af))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(2 * PI) * (1 / 2) is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * 0 is empty trivial complex real ext-real non positive non negative Function-like functional V67() V68() V69() V70() V71() V72() V73() Element of REAL
PI + ((2 * PI) * 0) is non empty complex real ext-real positive non negative Element of REAL
(3 / 2) - 1 is complex real ext-real Element of REAL
Af - 1 is complex real ext-real Element of REAL
(2 * PI) * (Af - 1) is complex real ext-real Element of REAL
(1 / 2) + 1 is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * ((1 / 2) + 1) is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
PI + ((2 * PI) * 1) is non empty complex real ext-real positive non negative Element of REAL
1 - Af is complex real ext-real Element of REAL
(AffineMap ((2 * PI),0)) . (1 - Af) is complex real ext-real Element of REAL
(2 * PI) * (1 - Af) is complex real ext-real Element of REAL
((2 * PI) * (1 - Af)) + 0 is complex real ext-real Element of REAL
((AffineMap ((2 * PI),0)) . (1 - Af)) / ((2 * PI) * 1) is complex real ext-real Element of REAL
(1 - Af) / 1 is complex real ext-real Element of REAL
(2 * PI) * (- 1) is non empty complex real ext-real non positive negative Element of REAL
((2 * PI) * Af) + ((2 * PI) * (- 1)) is complex real ext-real Element of REAL
cos (((2 * PI) * Af) + ((2 * PI) * (- 1))) is complex real ext-real Element of REAL
cos ((2 * PI) * (Af - 1)) is complex real ext-real Element of REAL
1 - 1 is complex real ext-real integer Element of REAL
((2 * PI) * (Af - 1)) / (2 * PI) is complex real ext-real Element of REAL
1 + (((2 * PI) * (Af - 1)) / (2 * PI)) is complex real ext-real Element of REAL
1 + (Af - 1) is complex real ext-real Element of REAL
(2 * PI) + ((2 * PI) * 0) is non empty complex real ext-real positive non negative Element of REAL
1 - (1 / 2) is complex real ext-real Element of REAL
1 - 1 is complex real ext-real integer Element of REAL
arccos (cos ((2 * PI) * Af)) is complex real ext-real Element of REAL
- ((2 * PI) * Af) is complex real ext-real Element of REAL
cos (- ((2 * PI) * Af)) is complex real ext-real Element of REAL
arccos (cos (- ((2 * PI) * Af))) is complex real ext-real Element of REAL
- Af is complex real ext-real Element of REAL
(2 * PI) * (- Af) is complex real ext-real Element of REAL
((2 * PI) * (- Af)) + ((2 * PI) * 1) is complex real ext-real Element of REAL
cos (((2 * PI) * (- Af)) + ((2 * PI) * 1)) is complex real ext-real Element of REAL
arccos (cos (((2 * PI) * (- Af)) + ((2 * PI) * 1))) is complex real ext-real Element of REAL
(() * j) . E is complex real ext-real set
(id the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) . E is complex real ext-real set
j " is V16() Function-like set
[.(- 1),1.[ is non empty V67() V68() V69() Element of K6(REAL)
].0,PI.] is non empty V67() V68() V69() Element of K6(REAL)
1 / (2 * PI) is non empty complex real ext-real positive non negative Element of REAL
Q is non empty V67() V68() V69() Element of K6(REAL)
(Q) is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | (Q) is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | (Q)) is non empty V67() V68() V69() set
E is non empty V67() V68() V69() Element of K6(REAL)
(E) is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | (E) is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | (E)) is non empty V67() V68() V69() set
AffineMap ((1 / (2 * PI)),0) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
(AffineMap ((1 / (2 * PI)),0)) | (E) is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (AffineMap ((1 / (2 * PI)),0)) is V67() V68() V69() Element of K6(REAL)
dom ((AffineMap ((1 / (2 * PI)),0)) | (E)) is V67() V68() V69() Element of K6(REAL)
rng ((AffineMap ((1 / (2 * PI)),0)) | (E)) is V67() V68() V69() Element of K6(REAL)
Af is set
Af is set
((AffineMap ((1 / (2 * PI)),0)) | (E)) . Af is complex real ext-real set
L is complex real ext-real Element of REAL
(AffineMap ((1 / (2 * PI)),0)) . L is complex real ext-real Element of REAL
(1 / (2 * PI)) * L is complex real ext-real Element of REAL
((1 / (2 * PI)) * L) + 0 is complex real ext-real Element of REAL
L / (2 * PI) is complex real ext-real Element of REAL
L is complex real ext-real Element of REAL
(1 * PI) / (2 * PI) is non empty complex real ext-real positive non negative Element of REAL
K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].0,1.[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].0,1.[)))) is non empty set
((AffineMap ((1 / (2 * PI)),0))) is V16() V19( the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),0)))))) V19( the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),0)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),0)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),0)))))) Function-like one-to-one quasi_total quasi_total onto onto bijective bijective V51() V52() V53() continuous continuous open Element of K6(K7( the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),0))))), the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),0)))))))
((dom (AffineMap ((1 / (2 * PI)),0)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom (AffineMap ((1 / (2 * PI)),0)))) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),0))))) is V67() V68() V69() set
rng (AffineMap ((1 / (2 * PI)),0)) is V67() V68() V69() Element of K6(REAL)
((rng (AffineMap ((1 / (2 * PI)),0)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng (AffineMap ((1 / (2 * PI)),0)))) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),0))))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),0))))), the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),0)))))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),0))))), the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),0))))))) is non empty set
R^1 | ([#] R^1) is non empty strict TopSpace-like V299() SubSpace of R^1
Af is non empty V16() V19( the carrier of (R^1 | (E))) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].0,1.[))))
AffineMap ((- 1),1) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (AffineMap ((- 1),1)) is V67() V68() V69() Element of K6(REAL)
((dom (AffineMap ((- 1),1)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom (AffineMap ((- 1),1)))) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom (AffineMap ((- 1),1))))) is V67() V68() V69() set
rng (AffineMap ((- 1),1)) is V67() V68() V69() Element of K6(REAL)
((rng (AffineMap ((- 1),1)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng (AffineMap ((- 1),1)))) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng (AffineMap ((- 1),1))))) is V67() V68() V69() set
((AffineMap ((- 1),1))) is V16() V19( the carrier of (R^1 | ((dom (AffineMap ((- 1),1)))))) V19( the carrier of (R^1 | ((dom (AffineMap ((- 1),1)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((- 1),1)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((- 1),1)))))) Function-like one-to-one quasi_total quasi_total onto onto bijective bijective V51() V52() V53() continuous continuous open Element of K6(K7( the carrier of (R^1 | ((dom (AffineMap ((- 1),1))))), the carrier of (R^1 | ((rng (AffineMap ((- 1),1)))))))
K7( the carrier of (R^1 | ((dom (AffineMap ((- 1),1))))), the carrier of (R^1 | ((rng (AffineMap ((- 1),1)))))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom (AffineMap ((- 1),1))))), the carrier of (R^1 | ((rng (AffineMap ((- 1),1))))))) is non empty set
((AffineMap ((- 1),1))) | (].0,1.[) is V16() V19( the carrier of (R^1 | ((dom (AffineMap ((- 1),1)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((- 1),1)))))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (R^1 | ((dom (AffineMap ((- 1),1))))), the carrier of (R^1 | ((rng (AffineMap ((- 1),1)))))))
dom (((AffineMap ((- 1),1))) | (].0,1.[)) is V67() V68() V69() Element of K6( the carrier of (R^1 | ((dom (AffineMap ((- 1),1))))))
K6( the carrier of (R^1 | ((dom (AffineMap ((- 1),1)))))) is non empty set
rng (((AffineMap ((- 1),1))) | (].0,1.[)) is V67() V68() V69() Element of K6( the carrier of (R^1 | ((rng (AffineMap ((- 1),1))))))
K6( the carrier of (R^1 | ((rng (AffineMap ((- 1),1)))))) is non empty set
L is set
L is set
(((AffineMap ((- 1),1))) | (].0,1.[)) . L is complex real ext-real set
ac is complex real ext-real Element of REAL
1 - ac is complex real ext-real Element of REAL
1 - 1 is complex real ext-real integer Element of REAL
(AffineMap ((- 1),1)) . ac is complex real ext-real Element of REAL
(- 1) * ac is complex real ext-real Element of REAL
((- 1) * ac) + 1 is complex real ext-real Element of REAL
K7( the carrier of (R^1 | (].0,1.[)), the carrier of (R^1 | (].0,1.[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | (].0,1.[)), the carrier of (R^1 | (].0,1.[)))) is non empty set
L is non empty V16() V19( the carrier of (R^1 | (].0,1.[))) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | (].0,1.[)), the carrier of (R^1 | (].0,1.[))))
([.(- 1),1.]) is non empty V67() V68() V69() closed Element of K6( the carrier of R^1)
R^1 | ([.(- 1),1.]) is non empty strict TopSpace-like closed V299() SubSpace of R^1
the carrier of (R^1 | ([.(- 1),1.])) is non empty V67() V68() V69() set
([.0,PI.]) is non empty V67() V68() V69() closed Element of K6( the carrier of R^1)
R^1 | ([.0,PI.]) is non empty strict TopSpace-like closed V299() SubSpace of R^1
the carrier of (R^1 | ([.0,PI.])) is non empty V67() V68() V69() set
K7( the carrier of (R^1 | ([.(- 1),1.])), the carrier of (R^1 | ([.0,PI.]))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | ([.(- 1),1.])), the carrier of (R^1 | ([.0,PI.])))) is non empty set
(arccos) is V16() V19( the carrier of (R^1 | ((dom arccos)))) V20( the carrier of (R^1 | ((rng arccos)))) Function-like one-to-one quasi_total onto bijective V51() V52() V53() continuous Element of K6(K7( the carrier of (R^1 | ((dom arccos))), the carrier of (R^1 | ((rng arccos)))))
((dom arccos)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom arccos)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom arccos))) is V67() V68() V69() set
((rng arccos)) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng arccos)) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng arccos))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom arccos))), the carrier of (R^1 | ((rng arccos)))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom arccos))), the carrier of (R^1 | ((rng arccos))))) is non empty set
ac is non empty V16() V19( the carrier of (R^1 | ([.(- 1),1.]))) V20( the carrier of (R^1 | ([.0,PI.]))) Function-like total quasi_total V51() V52() V53() continuous Element of K6(K7( the carrier of (R^1 | ([.(- 1),1.])), the carrier of (R^1 | ([.0,PI.]))))
ac | (Q) is V16() V19( the carrier of (R^1 | ([.(- 1),1.]))) V20( the carrier of (R^1 | ([.0,PI.]))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (R^1 | ([.(- 1),1.])), the carrier of (R^1 | ([.0,PI.]))))
dom (ac | (Q)) is V67() V68() V69() Element of K6( the carrier of (R^1 | ([.(- 1),1.])))
K6( the carrier of (R^1 | ([.(- 1),1.]))) is non empty set
rng (ac | (Q)) is V67() V68() V69() Element of K6( the carrier of (R^1 | ([.0,PI.])))
K6( the carrier of (R^1 | ([.0,PI.]))) is non empty set
c is set
QQ is set
(ac | (Q)) . QQ is complex real ext-real set
EE is complex real ext-real Element of REAL
rng ac is V67() V68() V69() Element of K6( the carrier of (R^1 | ([.0,PI.])))
p is complex real ext-real Element of REAL
arccos . EE is complex real ext-real set
arccos EE is complex real ext-real Element of REAL
K7( the carrier of (R^1 | (Q)), the carrier of (R^1 | (E))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | (Q)), the carrier of (R^1 | (E)))) is non empty set
QQ is V67() V68() V69() Element of K6( the carrier of (R^1 | ([.(- 1),1.])))
(R^1 | ([.(- 1),1.])) | QQ is strict TopSpace-like V299() SubSpace of R^1 | ([.(- 1),1.])
EE is V67() V68() V69() Element of K6( the carrier of (R^1 | ([.0,PI.])))
(R^1 | ([.0,PI.])) | EE is strict TopSpace-like V299() SubSpace of R^1 | ([.0,PI.])
c is non empty V16() V19( the carrier of (R^1 | (Q))) V20( the carrier of (R^1 | (E))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | (Q)), the carrier of (R^1 | (E))))
K7( the carrier of (TOP-REAL 2), the carrier of R^1) is non empty V51() V52() V53() set
K6(K7( the carrier of (TOP-REAL 2), the carrier of R^1)) is non empty set
p is non empty V16() V19( the carrier of (TOP-REAL 2)) V20( the carrier of R^1) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (TOP-REAL 2), the carrier of R^1))
dom p is functional Element of K6( the carrier of (TOP-REAL 2))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : ( b₁ in the carrier of (()) & 0 <= b₁ `2 ) } is set
h1 is Element of K6( the carrier of (()))
(()) | h1 is strict TopSpace-like V221() V222() SubSpace of (())
() | ((()) | h1) is V16() V19( the carrier of ((()) | h1)) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | h1), the carrier of (R^1 | (].0,1.[))))
the carrier of ((()) | h1) is set
K7( the carrier of ((()) | h1), the carrier of (R^1 | (].0,1.[))) is V51() V52() V53() set
K6(K7( the carrier of ((()) | h1), the carrier of (R^1 | (].0,1.[)))) is non empty set
() | the carrier of ((()) | h1) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].0,1.[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].0,1.[))))
h1 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
h1 `2 is complex real ext-real Element of REAL
A is non empty Element of K6( the carrier of (()))
(()) | A is non empty strict TopSpace-like V221() V222() SubSpace of (())
[#] ((()) | A) is non empty non proper open closed dense non boundary Element of K6( the carrier of ((()) | A))
the carrier of ((()) | A) is non empty set
K6( the carrier of ((()) | A)) is non empty set
E is non empty functional Element of K6( the carrier of (TOP-REAL 2))
p | E is V16() V19( the carrier of (TOP-REAL 2)) V20( the carrier of R^1) Function-like V51() V52() V53() Element of K6(K7( the carrier of (TOP-REAL 2), the carrier of R^1))
dom (p | E) is functional Element of K6( the carrier of (TOP-REAL 2))
rng (p | E) is V67() V68() V69() Element of K6( the carrier of R^1)
E is set
Af is set
(p | E) . Af is complex real ext-real set
Af is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Af `2 is complex real ext-real Element of REAL
Af `1 is complex real ext-real Element of REAL
p . Af is complex real ext-real set
(TOP-REAL 2) | E is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
the carrier of ((TOP-REAL 2) | E) is non empty set
K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (Q))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (Q)))) is non empty set
() | ((()) | A) is non empty V16() V19( the carrier of ((()) | A)) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | A), the carrier of (R^1 | (].0,1.[))))
K7( the carrier of ((()) | A), the carrier of (R^1 | (].0,1.[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((()) | A), the carrier of (R^1 | (].0,1.[)))) is non empty set
() | the carrier of ((()) | A) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].0,1.[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].0,1.[))))
E is non empty V16() V19( the carrier of ((TOP-REAL 2) | E)) V20( the carrier of (R^1 | (Q))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (Q))))
c * E is non empty V16() V19( the carrier of ((TOP-REAL 2) | E)) V20( the carrier of (R^1 | (E))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (E))))
K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (E))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (E)))) is non empty set
Af is non empty V16() V19( the carrier of (R^1 | (E))) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() continuous Element of K6(K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].0,1.[))))
Af * (c * E) is non empty V16() V19( the carrier of ((TOP-REAL 2) | E)) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (].0,1.[))))
K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (].0,1.[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (].0,1.[)))) is non empty set
Af is V16() Function-like FinSequence-like Element of the carrier of ((()) | A)
(() | ((()) | A)) . Af is complex real ext-real Element of the carrier of (R^1 | (].0,1.[))
(Af * (c * E)) . Af is complex real ext-real set
Af is V16() Function-like FinSequence-like Element of the carrier of (())
() . Af is complex real ext-real Element of the carrier of (R^1 | (].0,1.[))
Af is complex real ext-real set
Af1 is complex real ext-real set
|[Af,Af1]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos Af is complex real ext-real Element of REAL
(arccos Af) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos Af) / (2 * PI)) is complex real ext-real Element of REAL
|[Af,Af1]| `1 is complex real ext-real Element of REAL
arccos . Af is complex real ext-real set
c . Af is complex real ext-real set
rng c is V67() V68() V69() Element of K6( the carrier of (R^1 | (E)))
K6( the carrier of (R^1 | (E))) is non empty set
(arccos . Af) / (2 * PI) is complex real ext-real Element of REAL
(1 / (2 * PI)) * (arccos . Af) is complex real ext-real Element of REAL
((1 / (2 * PI)) * (arccos . Af)) + 0 is complex real ext-real Element of REAL
(AffineMap ((1 / (2 * PI)),0)) . (arccos . Af) is complex real ext-real set
Af . (arccos . Af) is complex real ext-real set
Af . (c . Af) is complex real ext-real set
c . (|[Af,Af1]| `1) is complex real ext-real set
Af . (c . (|[Af,Af1]| `1)) is complex real ext-real set
p . Af is complex real ext-real set
c . (p . Af) is complex real ext-real set
Af . (c . (p . Af)) is complex real ext-real set
E . Af is complex real ext-real set
c . (E . Af) is complex real ext-real set
Af . (c . (E . Af)) is complex real ext-real set
(c * E) . Af is complex real ext-real set
Af . ((c * E) . Af) is complex real ext-real set
Af1 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Af1 `2 is complex real ext-real Element of REAL
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : ( b₁ in the carrier of (()) & b₁ `2 <= 0 ) } is set
h1 is Element of K6( the carrier of (()))
(()) | h1 is strict TopSpace-like V221() V222() SubSpace of (())
() | ((()) | h1) is V16() V19( the carrier of ((()) | h1)) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | h1), the carrier of (R^1 | (].0,1.[))))
the carrier of ((()) | h1) is set
K7( the carrier of ((()) | h1), the carrier of (R^1 | (].0,1.[))) is V51() V52() V53() set
K6(K7( the carrier of ((()) | h1), the carrier of (R^1 | (].0,1.[)))) is non empty set
() | the carrier of ((()) | h1) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].0,1.[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].0,1.[))))
h1 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
h1 `2 is complex real ext-real Element of REAL
A is non empty Element of K6( the carrier of (()))
(()) | A is non empty strict TopSpace-like V221() V222() SubSpace of (())
[#] ((()) | A) is non empty non proper open closed dense non boundary Element of K6( the carrier of ((()) | A))
the carrier of ((()) | A) is non empty set
K6( the carrier of ((()) | A)) is non empty set
E is non empty functional Element of K6( the carrier of (TOP-REAL 2))
p | E is V16() V19( the carrier of (TOP-REAL 2)) V20( the carrier of R^1) Function-like V51() V52() V53() Element of K6(K7( the carrier of (TOP-REAL 2), the carrier of R^1))
dom (p | E) is functional Element of K6( the carrier of (TOP-REAL 2))
rng (p | E) is V67() V68() V69() Element of K6( the carrier of R^1)
E is set
Af is set
(p | E) . Af is complex real ext-real set
Af is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Af `2 is complex real ext-real Element of REAL
Af `1 is complex real ext-real Element of REAL
p . Af is complex real ext-real set
(TOP-REAL 2) | E is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
the carrier of ((TOP-REAL 2) | E) is non empty set
K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (Q))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (Q)))) is non empty set
() | ((()) | A) is non empty V16() V19( the carrier of ((()) | A)) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | A), the carrier of (R^1 | (].0,1.[))))
K7( the carrier of ((()) | A), the carrier of (R^1 | (].0,1.[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((()) | A), the carrier of (R^1 | (].0,1.[)))) is non empty set
() | the carrier of ((()) | A) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].0,1.[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].0,1.[))))
E is non empty V16() V19( the carrier of ((TOP-REAL 2) | E)) V20( the carrier of (R^1 | (Q))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (Q))))
c * E is non empty V16() V19( the carrier of ((TOP-REAL 2) | E)) V20( the carrier of (R^1 | (E))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (E))))
K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (E))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (E)))) is non empty set
Af is non empty V16() V19( the carrier of (R^1 | (E))) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() continuous Element of K6(K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].0,1.[))))
Af * (c * E) is non empty V16() V19( the carrier of ((TOP-REAL 2) | E)) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (].0,1.[))))
K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (].0,1.[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (].0,1.[)))) is non empty set
L is non empty V16() V19( the carrier of (R^1 | (].0,1.[))) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() continuous Element of K6(K7( the carrier of (R^1 | (].0,1.[)), the carrier of (R^1 | (].0,1.[))))
L * (Af * (c * E)) is non empty V16() V19( the carrier of ((TOP-REAL 2) | E)) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | E), the carrier of (R^1 | (].0,1.[))))
Af is V16() Function-like FinSequence-like Element of the carrier of ((()) | A)
(() | ((()) | A)) . Af is complex real ext-real Element of the carrier of (R^1 | (].0,1.[))
(L * (Af * (c * E))) . Af is complex real ext-real set
Af is V16() Function-like FinSequence-like Element of the carrier of (())
() . Af is complex real ext-real Element of the carrier of (R^1 | (].0,1.[))
Af is complex real ext-real set
Af1 is complex real ext-real set
|[Af,Af1]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos Af is complex real ext-real Element of REAL
(arccos Af) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos Af) / (2 * PI)) is complex real ext-real Element of REAL
|[Af,Af1]| `1 is complex real ext-real Element of REAL
dom (Af * (c * E)) is Element of K6( the carrier of ((TOP-REAL 2) | E))
K6( the carrier of ((TOP-REAL 2) | E)) is non empty set
(Af * (c * E)) . Af is complex real ext-real set
rng (Af * (c * E)) is V67() V68() V69() Element of K6( the carrier of (R^1 | (].0,1.[)))
K6( the carrier of (R^1 | (].0,1.[))) is non empty set
arccos . Af is complex real ext-real set
c . Af is complex real ext-real set
rng c is V67() V68() V69() Element of K6( the carrier of (R^1 | (E)))
K6( the carrier of (R^1 | (E))) is non empty set
(arccos . Af) / (2 * PI) is complex real ext-real Element of REAL
(1 / (2 * PI)) * (arccos . Af) is complex real ext-real Element of REAL
((1 / (2 * PI)) * (arccos . Af)) + 0 is complex real ext-real Element of REAL
(AffineMap ((1 / (2 * PI)),0)) . (arccos . Af) is complex real ext-real set
Af . (arccos . Af) is complex real ext-real set
Af . (c . Af) is complex real ext-real set
c . (|[Af,Af1]| `1) is complex real ext-real set
Af . (c . (|[Af,Af1]| `1)) is complex real ext-real set
p . Af is complex real ext-real set
c . (p . Af) is complex real ext-real set
Af . (c . (p . Af)) is complex real ext-real set
E . Af is complex real ext-real set
c . (E . Af) is complex real ext-real set
Af . (c . (E . Af)) is complex real ext-real set
(c * E) . Af is complex real ext-real set
Af . ((c * E) . Af) is complex real ext-real set
Af1 is complex real ext-real Element of REAL
(- 1) * Af1 is complex real ext-real Element of REAL
((- 1) * Af1) + 1 is complex real ext-real Element of REAL
((AffineMap ((- 1),1))) . Af1 is complex real ext-real set
L . Af1 is complex real ext-real set
Af1 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Af1 `2 is complex real ext-real Element of REAL
Q is set
E is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
E `2 is complex real ext-real Element of REAL
Q is set
E is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
E `2 is complex real ext-real Element of REAL
h1 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
h1 `2 is complex real ext-real Element of REAL
Q is non empty Element of K6( the carrier of (()))
(()) | Q is non empty strict TopSpace-like V221() V222() SubSpace of (())
E is V16() Function-like FinSequence-like Element of the carrier of (())
the carrier of ((()) | Q) is non empty set
E is non empty Element of K6( the carrier of (()))
(()) | E is non empty strict TopSpace-like V221() V222() SubSpace of (())
the carrier of ((()) | E) is non empty set
Q \/ E is non empty Element of K6( the carrier of (()))
Af1 is set
Af1 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Af1 `2 is complex real ext-real Element of REAL
() | ((()) | E) is non empty V16() V19( the carrier of ((()) | E)) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | E), the carrier of (R^1 | (].0,1.[))))
K7( the carrier of ((()) | E), the carrier of (R^1 | (].0,1.[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((()) | E), the carrier of (R^1 | (].0,1.[)))) is non empty set
() | the carrier of ((()) | E) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].0,1.[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].0,1.[))))
Af is V16() Function-like FinSequence-like Element of the carrier of ((()) | E)
() | ((()) | Q) is non empty V16() V19( the carrier of ((()) | Q)) V20( the carrier of (R^1 | (].0,1.[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | Q), the carrier of (R^1 | (].0,1.[))))
K7( the carrier of ((()) | Q), the carrier of (R^1 | (].0,1.[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((()) | Q), the carrier of (R^1 | (].0,1.[)))) is non empty set
() | the carrier of ((()) | Q) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].0,1.[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].0,1.[))))
Af is V16() Function-like FinSequence-like Element of the carrier of ((()) | Q)
((()) | Q) union ((()) | E) is non empty strict TopSpace-like V221() V222() SubSpace of (())
REAL --> 1 is non empty V16() non-empty V19( REAL ) V20( REAL ) V20( RAT ) V20( INT ) Function-like constant total quasi_total continuous Lipschitzian V51() V52() V53() V54() Element of K6(K7(REAL,REAL))
(1 / (2 * PI)) (#) arccos is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
[.p0,0.[ is non empty V67() V68() V69() Element of K6(REAL)
].0,p1.] is non empty V67() V68() V69() Element of K6(REAL)
Af is V16() Function-like FinSequence-like Element of the carrier of (())
[#] (Closed-Interval-TSpace ((- 1),1)) is non empty non proper V67() V68() V69() open closed dense non boundary Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
K6( the carrier of (Closed-Interval-TSpace ((- 1),1))) is non empty set
() . Af is complex real ext-real Element of the carrier of (R^1 | (].0,1.[))
Af is complex real ext-real set
Af1 is complex real ext-real set
|[Af,Af1]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos Af is complex real ext-real Element of REAL
(arccos Af) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos Af) / (2 * PI)) is complex real ext-real Element of REAL
Af is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
Af `2 is complex real ext-real Element of REAL
Af `1 is complex real ext-real Element of REAL
((1 / (2 * PI)) (#) arccos) | [.(- 1),1.] is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
(((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) . Af is complex real ext-real set
((1 / (2 * PI)) (#) arccos) . Af is complex real ext-real set
dom ((1 / (2 * PI)) (#) arccos) is V67() V68() V69() Element of K6(REAL)
dom (((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) is V67() V68() V69() Element of K6(REAL)
arccos . Af is complex real ext-real set
(arccos . Af) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 * (arccos . Af) is complex real ext-real Element of REAL
(1 * (arccos . Af)) / (2 * PI) is complex real ext-real Element of REAL
K6( the carrier of (R^1 | (].0,(0 + p1).[))) is non empty set
h1 is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
h1 - ((1 / (2 * PI)) (#) arccos) is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
Af1 is V67() V68() V69() Element of K6( the carrier of (R^1 | (].0,(0 + p1).[)))
c is V67() V68() V69() Element of K6(REAL)
c is V67() V68() V69() Element of K6( the carrier of R^1)
QQ is open V67() V68() V69() Element of K6(REAL)
EE is open V67() V68() V69() Neighbourhood of () . Af
(h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.] is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . Af is complex real ext-real set
(h1 - ((1 / (2 * PI)) (#) arccos)) . Af is complex real ext-real set
dom (h1 - ((1 / (2 * PI)) (#) arccos)) is V67() V68() V69() Element of K6(REAL)
dom h1 is V67() V68() V69() Element of K6(REAL)
(dom h1) /\ (dom ((1 / (2 * PI)) (#) arccos)) is V67() V68() V69() Element of K6(REAL)
REAL /\ (dom ((1 / (2 * PI)) (#) arccos)) is V67() V68() V69() Element of K6(REAL)
dom ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) is V67() V68() V69() Element of K6(REAL)
(arccos . Af) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos . Af) / (2 * PI)) is complex real ext-real Element of REAL
(1,2) --> (Af,Af1) is set
h1 . Af is complex real ext-real set
(h1 . Af) - (((1 / (2 * PI)) (#) arccos) . Af) is complex real ext-real set
1 - ((1 * (arccos . Af)) / (2 * PI)) is complex real ext-real Element of REAL
D2 is open V67() V68() V69() Neighbourhood of Af
((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: D2 is V67() V68() V69() Element of K6(REAL)
D2 /\ [.(- 1),1.] is V67() V68() V69() Element of K6(REAL)
F1 is V67() V68() V69() Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
F2 is V67() V68() V69() Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
(1,2) --> (F1,F2) is non empty V16() V19({1,2}) V20(K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) Function-like total quasi_total Element of K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))))
{1,2} is non empty V67() V68() V69() V70() V71() V72() set
K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) is non empty set
K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1))))) is non empty set
product ((1,2) --> (F1,F2)) is set
(product ((1,2) --> (F1,F2))) /\ the carrier of (()) is set
F is Element of K6( the carrier of (()))
() .: F is V67() V68() V69() Element of K6( the carrier of (R^1 | (].0,1.[)))
K6( the carrier of (R^1 | (].0,1.[))) is non empty set
the carrier of (Trectangle (p0,p1,p0,p1)) is non empty set
K6( the carrier of (Trectangle (p0,p1,p0,p1))) is non empty set
((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: F1 is V67() V68() V69() Element of K6(REAL)
(D2) is V67() V68() V69() open Element of K6( the carrier of R^1)
U is Element of K6( the carrier of (Trectangle (p0,p1,p0,p1)))
d is V67() V68() V69() open Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
U1 is V67() V68() V69() open Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
(1,2) --> (d,U1) is non empty V16() V19({1,2}) V20(K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) Function-like total quasi_total Element of K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))))
product ((1,2) --> (d,U1)) is set
n is set
d1 is V16() Function-like FinSequence-like Element of the carrier of (())
() . d1 is complex real ext-real Element of the carrier of (R^1 | (].0,1.[))
d1 . 1 is set
f is complex real ext-real set
f1 is complex real ext-real set
|[f,f1]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos f is complex real ext-real Element of REAL
(arccos f) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos f) / (2 * PI)) is complex real ext-real Element of REAL
d1 . 2 is set
1 * (arccos f) is complex real ext-real Element of REAL
(1 * (arccos f)) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 - ((1 * (arccos f)) * (1 / (2 * PI))) is complex real ext-real Element of REAL
((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . (d1 . 1) is complex real ext-real set
(h1 - ((1 / (2 * PI)) (#) arccos)) . (d1 . 1) is complex real ext-real set
h1 . (d1 . 1) is complex real ext-real set
((1 / (2 * PI)) (#) arccos) . (d1 . 1) is complex real ext-real set
(h1 . (d1 . 1)) - (((1 / (2 * PI)) (#) arccos) . (d1 . 1)) is complex real ext-real set
1 - (((1 / (2 * PI)) (#) arccos) . (d1 . 1)) is complex real ext-real Element of REAL
arccos . (d1 . 1) is complex real ext-real set
(arccos . (d1 . 1)) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos . (d1 . 1)) * (1 / (2 * PI))) is complex real ext-real Element of REAL
arccos . f is complex real ext-real set
(arccos . f) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos . f) * (1 / (2 * PI))) is complex real ext-real Element of REAL
(arccos f) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos f) * (1 / (2 * PI))) is complex real ext-real Element of REAL
K6( the carrier of (R^1 | (].0,(0 + p1).[))) is non empty set
Af1 is V67() V68() V69() Element of K6( the carrier of (R^1 | (].0,(0 + p1).[)))
Af1 is V67() V68() V69() Element of K6(REAL)
c is V67() V68() V69() Element of K6( the carrier of R^1)
c is open V67() V68() V69() Element of K6(REAL)
QQ is open V67() V68() V69() Neighbourhood of () . Af
(arccos . Af) / (2 * PI) is complex real ext-real Element of REAL
EE is open V67() V68() V69() Neighbourhood of Af
(((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) .: EE is V67() V68() V69() Element of K6(REAL)
EE /\ [.(- 1),1.] is V67() V68() V69() Element of K6(REAL)
D1 is V67() V68() V69() Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
F1 is V67() V68() V69() Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
(1,2) --> (D1,F1) is non empty V16() V19({1,2}) V20(K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) Function-like total quasi_total Element of K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))))
{1,2} is non empty V67() V68() V69() V70() V71() V72() set
K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) is non empty set
K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1))))) is non empty set
product ((1,2) --> (D1,F1)) is set
(product ((1,2) --> (D1,F1))) /\ the carrier of (()) is set
F is Element of K6( the carrier of (()))
() .: F is V67() V68() V69() Element of K6( the carrier of (R^1 | (].0,1.[)))
K6( the carrier of (R^1 | (].0,1.[))) is non empty set
the carrier of (Trectangle (p0,p1,p0,p1)) is non empty set
K6( the carrier of (Trectangle (p0,p1,p0,p1))) is non empty set
(((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) .: D1 is V67() V68() V69() Element of K6(REAL)
(EE) is V67() V68() V69() open Element of K6( the carrier of R^1)
F is Element of K6( the carrier of (Trectangle (p0,p1,p0,p1)))
U1 is V67() V68() V69() open Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
U is V67() V68() V69() open Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
(1,2) --> (U1,U) is non empty V16() V19({1,2}) V20(K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) Function-like total quasi_total Element of K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))))
product ((1,2) --> (U1,U)) is set
(1,2) --> (Af,Af1) is set
d is set
n is V16() Function-like FinSequence-like Element of the carrier of (())
() . n is complex real ext-real Element of the carrier of (R^1 | (].0,1.[))
n . 1 is set
d1 is complex real ext-real set
f is complex real ext-real set
|[d1,f]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos d1 is complex real ext-real Element of REAL
(arccos d1) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos d1) / (2 * PI)) is complex real ext-real Element of REAL
n . 2 is set
1 * (arccos d1) is complex real ext-real Element of REAL
(1 * (arccos d1)) * (1 / (2 * PI)) is complex real ext-real Element of REAL
(((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) . (n . 1) is complex real ext-real set
((1 / (2 * PI)) (#) arccos) . (n . 1) is complex real ext-real set
arccos . (n . 1) is complex real ext-real set
(arccos . (n . 1)) * (1 / (2 * PI)) is complex real ext-real Element of REAL
arccos . d1 is complex real ext-real set
(arccos . d1) * (1 / (2 * PI)) is complex real ext-real Element of REAL
(arccos d1) * (1 / (2 * PI)) is complex real ext-real Element of REAL
].(- 1),1.] is non empty V67() V68() V69() Element of K6(REAL)
[.0,PI.[ is non empty V67() V68() V69() Element of K6(REAL)
Q is non empty V67() V68() V69() Element of K6(REAL)
(Q) is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | (Q) is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | (Q)) is non empty V67() V68() V69() set
E is non empty V67() V68() V69() Element of K6(REAL)
(E) is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | (E) is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | (E)) is non empty V67() V68() V69() set
- (1 / (2 * PI)) is non empty complex real ext-real non positive negative Element of REAL
AffineMap ((- (1 / (2 * PI))),1) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
(AffineMap ((- (1 / (2 * PI))),1)) | (E) is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (AffineMap ((- (1 / (2 * PI))),1)) is V67() V68() V69() Element of K6(REAL)
dom ((AffineMap ((- (1 / (2 * PI))),1)) | (E)) is V67() V68() V69() Element of K6(REAL)
rng ((AffineMap ((- (1 / (2 * PI))),1)) | (E)) is V67() V68() V69() Element of K6(REAL)
Af is set
Af is set
((AffineMap ((- (1 / (2 * PI))),1)) | (E)) . Af is complex real ext-real set
Af1 is complex real ext-real Element of REAL
(AffineMap ((- (1 / (2 * PI))),1)) . Af1 is complex real ext-real Element of REAL
(- (1 / (2 * PI))) * Af1 is complex real ext-real Element of REAL
((- (1 / (2 * PI))) * Af1) + 1 is complex real ext-real Element of REAL
(1 / (2 * PI)) * Af1 is complex real ext-real Element of REAL
- ((1 / (2 * PI)) * Af1) is complex real ext-real Element of REAL
(- ((1 / (2 * PI)) * Af1)) + 1 is complex real ext-real Element of REAL
Af1 / (2 * PI) is complex real ext-real Element of REAL
- (Af1 / (2 * PI)) is complex real ext-real Element of REAL
(- (Af1 / (2 * PI))) + 1 is complex real ext-real Element of REAL
(1 * PI) / (2 * PI) is non empty complex real ext-real positive non negative Element of REAL
- (1 / 2) is non empty complex real ext-real non positive negative Element of REAL
(- (1 / 2)) + 1 is complex real ext-real Element of REAL
0 + 1 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
Af1 is complex real ext-real Element of REAL
K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)))) is non empty set
((AffineMap ((- (1 / (2 * PI))),1))) is V16() V19( the carrier of (R^1 | ((dom (AffineMap ((- (1 / (2 * PI))),1)))))) V19( the carrier of (R^1 | ((dom (AffineMap ((- (1 / (2 * PI))),1)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((- (1 / (2 * PI))),1)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((- (1 / (2 * PI))),1)))))) Function-like one-to-one quasi_total quasi_total onto onto bijective bijective V51() V52() V53() continuous continuous open Element of K6(K7( the carrier of (R^1 | ((dom (AffineMap ((- (1 / (2 * PI))),1))))), the carrier of (R^1 | ((rng (AffineMap ((- (1 / (2 * PI))),1)))))))
((dom (AffineMap ((- (1 / (2 * PI))),1)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom (AffineMap ((- (1 / (2 * PI))),1)))) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom (AffineMap ((- (1 / (2 * PI))),1))))) is V67() V68() V69() set
rng (AffineMap ((- (1 / (2 * PI))),1)) is V67() V68() V69() Element of K6(REAL)
((rng (AffineMap ((- (1 / (2 * PI))),1)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng (AffineMap ((- (1 / (2 * PI))),1)))) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng (AffineMap ((- (1 / (2 * PI))),1))))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom (AffineMap ((- (1 / (2 * PI))),1))))), the carrier of (R^1 | ((rng (AffineMap ((- (1 / (2 * PI))),1)))))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom (AffineMap ((- (1 / (2 * PI))),1))))), the carrier of (R^1 | ((rng (AffineMap ((- (1 / (2 * PI))),1))))))) is non empty set
Af is non empty V16() V19( the carrier of (R^1 | (E))) V20( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))))
AffineMap ((1 / (2 * PI)),1) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
(AffineMap ((1 / (2 * PI)),1)) | (E) is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (AffineMap ((1 / (2 * PI)),1)) is V67() V68() V69() Element of K6(REAL)
dom ((AffineMap ((1 / (2 * PI)),1)) | (E)) is V67() V68() V69() Element of K6(REAL)
rng ((AffineMap ((1 / (2 * PI)),1)) | (E)) is V67() V68() V69() Element of K6(REAL)
Af1 is set
Af1 is set
((AffineMap ((1 / (2 * PI)),1)) | (E)) . Af1 is complex real ext-real set
c is complex real ext-real Element of REAL
(AffineMap ((1 / (2 * PI)),1)) . c is complex real ext-real Element of REAL
(1 / (2 * PI)) * c is complex real ext-real Element of REAL
((1 / (2 * PI)) * c) + 1 is complex real ext-real Element of REAL
c / (2 * PI) is complex real ext-real Element of REAL
(c / (2 * PI)) + 1 is complex real ext-real Element of REAL
(1 * PI) / (2 * PI) is non empty complex real ext-real positive non negative Element of REAL
(1 / 2) + 1 is non empty complex real ext-real positive non negative Element of REAL
0 + 1 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
c is complex real ext-real Element of REAL
((AffineMap ((1 / (2 * PI)),1))) is V16() V19( the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),1)))))) V19( the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),1)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),1)))))) V20( the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),1)))))) Function-like one-to-one quasi_total quasi_total onto onto bijective bijective V51() V52() V53() continuous continuous open Element of K6(K7( the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),1))))), the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),1)))))))
((dom (AffineMap ((1 / (2 * PI)),1)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom (AffineMap ((1 / (2 * PI)),1)))) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),1))))) is V67() V68() V69() set
rng (AffineMap ((1 / (2 * PI)),1)) is V67() V68() V69() Element of K6(REAL)
((rng (AffineMap ((1 / (2 * PI)),1)))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng (AffineMap ((1 / (2 * PI)),1)))) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),1))))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),1))))), the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),1)))))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom (AffineMap ((1 / (2 * PI)),1))))), the carrier of (R^1 | ((rng (AffineMap ((1 / (2 * PI)),1))))))) is non empty set
Af1 is non empty V16() V19( the carrier of (R^1 | (E))) V20( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))))
ac | (Q) is V16() V19( the carrier of (R^1 | ([.(- 1),1.]))) V20( the carrier of (R^1 | ([.0,PI.]))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (R^1 | ([.(- 1),1.])), the carrier of (R^1 | ([.0,PI.]))))
dom (ac | (Q)) is V67() V68() V69() Element of K6( the carrier of (R^1 | ([.(- 1),1.])))
rng (ac | (Q)) is V67() V68() V69() Element of K6( the carrier of (R^1 | ([.0,PI.])))
c is set
QQ is set
(ac | (Q)) . QQ is complex real ext-real set
rng ac is V67() V68() V69() Element of K6( the carrier of (R^1 | ([.0,PI.])))
EE is complex real ext-real Element of REAL
D2 is complex real ext-real Element of REAL
arccos . D2 is complex real ext-real set
arccos D2 is complex real ext-real Element of REAL
K7( the carrier of (R^1 | (Q)), the carrier of (R^1 | (E))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | (Q)), the carrier of (R^1 | (E)))) is non empty set
QQ is V67() V68() V69() Element of K6( the carrier of (R^1 | ([.(- 1),1.])))
(R^1 | ([.(- 1),1.])) | QQ is strict TopSpace-like V299() SubSpace of R^1 | ([.(- 1),1.])
EE is V67() V68() V69() Element of K6( the carrier of (R^1 | ([.0,PI.])))
(R^1 | ([.0,PI.])) | EE is strict TopSpace-like V299() SubSpace of R^1 | ([.0,PI.])
c is non empty V16() V19( the carrier of (R^1 | (Q))) V20( the carrier of (R^1 | (E))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | (Q)), the carrier of (R^1 | (E))))
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : ( b₁ in the carrier of (()) & 0 <= b₁ `2 ) } is set
D1 is Element of K6( the carrier of (()))
(()) | D1 is strict TopSpace-like V221() V222() SubSpace of (())
() | ((()) | D1) is V16() V19( the carrier of ((()) | D1)) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | D1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
the carrier of ((()) | D1) is set
K7( the carrier of ((()) | D1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) is V51() V52() V53() set
K6(K7( the carrier of ((()) | D1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[)))) is non empty set
() | the carrier of ((()) | D1) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
D2 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
D2 `2 is complex real ext-real Element of REAL
F1 is non empty Element of K6( the carrier of (()))
(()) | F1 is non empty strict TopSpace-like V221() V222() SubSpace of (())
[#] ((()) | F1) is non empty non proper open closed dense non boundary Element of K6( the carrier of ((()) | F1))
the carrier of ((()) | F1) is non empty set
K6( the carrier of ((()) | F1)) is non empty set
F is non empty functional Element of K6( the carrier of (TOP-REAL 2))
p | F is V16() V19( the carrier of (TOP-REAL 2)) V20( the carrier of R^1) Function-like V51() V52() V53() Element of K6(K7( the carrier of (TOP-REAL 2), the carrier of R^1))
dom (p | F) is functional Element of K6( the carrier of (TOP-REAL 2))
rng (p | F) is V67() V68() V69() Element of K6( the carrier of R^1)
U is set
U1 is set
(p | F) . U1 is complex real ext-real set
d is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
d `2 is complex real ext-real Element of REAL
d `1 is complex real ext-real Element of REAL
p . U1 is complex real ext-real set
(TOP-REAL 2) | F is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
the carrier of ((TOP-REAL 2) | F) is non empty set
K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (Q))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (Q)))) is non empty set
() | ((()) | F1) is non empty V16() V19( the carrier of ((()) | F1)) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | F1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
K7( the carrier of ((()) | F1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((()) | F1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[)))) is non empty set
() | the carrier of ((()) | F1) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
U is non empty V16() V19( the carrier of ((TOP-REAL 2) | F)) V20( the carrier of (R^1 | (Q))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (Q))))
c * U is non empty V16() V19( the carrier of ((TOP-REAL 2) | F)) V20( the carrier of (R^1 | (E))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (E))))
K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (E))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (E)))) is non empty set
Af1 is non empty V16() V19( the carrier of (R^1 | (E))) V20( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) Function-like total quasi_total V51() V52() V53() continuous Element of K6(K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))))
Af1 * (c * U) is non empty V16() V19( the carrier of ((TOP-REAL 2) | F)) V20( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))))
K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)))) is non empty set
U1 is V16() Function-like FinSequence-like Element of the carrier of ((()) | F1)
(() | ((()) | F1)) . U1 is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),(3 / 2).[))
(Af1 * (c * U)) . U1 is complex real ext-real set
d is V16() Function-like FinSequence-like Element of the carrier of (())
() . d is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),(3 / 2).[))
n is complex real ext-real set
d1 is complex real ext-real set
|[n,d1]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos n is complex real ext-real Element of REAL
(arccos n) / (2 * PI) is complex real ext-real Element of REAL
1 + ((arccos n) / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos n) / (2 * PI)) is complex real ext-real Element of REAL
|[n,d1]| `1 is complex real ext-real Element of REAL
arccos . n is complex real ext-real set
c . n is complex real ext-real set
rng c is V67() V68() V69() Element of K6( the carrier of (R^1 | (E)))
K6( the carrier of (R^1 | (E))) is non empty set
(arccos . n) / (2 * PI) is complex real ext-real Element of REAL
1 + ((arccos . n) / (2 * PI)) is complex real ext-real Element of REAL
(1 / (2 * PI)) * (arccos . n) is complex real ext-real Element of REAL
1 + ((1 / (2 * PI)) * (arccos . n)) is complex real ext-real Element of REAL
(AffineMap ((1 / (2 * PI)),1)) . (arccos . n) is complex real ext-real set
Af1 . (arccos . n) is complex real ext-real set
Af1 . (c . n) is complex real ext-real set
c . (|[n,d1]| `1) is complex real ext-real set
Af1 . (c . (|[n,d1]| `1)) is complex real ext-real set
p . U1 is complex real ext-real set
c . (p . U1) is complex real ext-real set
Af1 . (c . (p . U1)) is complex real ext-real set
U . U1 is complex real ext-real set
c . (U . U1) is complex real ext-real set
Af1 . (c . (U . U1)) is complex real ext-real set
(c * U) . U1 is complex real ext-real set
Af1 . ((c * U) . U1) is complex real ext-real set
f is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
f `2 is complex real ext-real Element of REAL
{ b₁ where b₁ is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2) : ( b₁ in the carrier of (()) & b₁ `2 <= 0 ) } is set
D1 is Element of K6( the carrier of (()))
(()) | D1 is strict TopSpace-like V221() V222() SubSpace of (())
() | ((()) | D1) is V16() V19( the carrier of ((()) | D1)) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | D1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
the carrier of ((()) | D1) is set
K7( the carrier of ((()) | D1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) is V51() V52() V53() set
K6(K7( the carrier of ((()) | D1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[)))) is non empty set
() | the carrier of ((()) | D1) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
D2 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
D2 `2 is complex real ext-real Element of REAL
F1 is non empty Element of K6( the carrier of (()))
(()) | F1 is non empty strict TopSpace-like V221() V222() SubSpace of (())
[#] ((()) | F1) is non empty non proper open closed dense non boundary Element of K6( the carrier of ((()) | F1))
the carrier of ((()) | F1) is non empty set
K6( the carrier of ((()) | F1)) is non empty set
F is non empty functional Element of K6( the carrier of (TOP-REAL 2))
p | F is V16() V19( the carrier of (TOP-REAL 2)) V20( the carrier of R^1) Function-like V51() V52() V53() Element of K6(K7( the carrier of (TOP-REAL 2), the carrier of R^1))
dom (p | F) is functional Element of K6( the carrier of (TOP-REAL 2))
rng (p | F) is V67() V68() V69() Element of K6( the carrier of R^1)
U is set
U1 is set
(p | F) . U1 is complex real ext-real set
d is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
d `2 is complex real ext-real Element of REAL
d `1 is complex real ext-real Element of REAL
p . U1 is complex real ext-real set
(TOP-REAL 2) | F is non empty strict TopSpace-like V221() V222() SubSpace of TOP-REAL 2
the carrier of ((TOP-REAL 2) | F) is non empty set
K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (Q))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (Q)))) is non empty set
() | ((()) | F1) is non empty V16() V19( the carrier of ((()) | F1)) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | F1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
K7( the carrier of ((()) | F1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((()) | F1), the carrier of (R^1 | (].(1 / 2),(3 / 2).[)))) is non empty set
() | the carrier of ((()) | F1) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
U is non empty V16() V19( the carrier of ((TOP-REAL 2) | F)) V20( the carrier of (R^1 | (Q))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (Q))))
c * U is non empty V16() V19( the carrier of ((TOP-REAL 2) | F)) V20( the carrier of (R^1 | (E))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (E))))
K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (E))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (E)))) is non empty set
Af is non empty V16() V19( the carrier of (R^1 | (E))) V20( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) Function-like total quasi_total V51() V52() V53() continuous Element of K6(K7( the carrier of (R^1 | (E)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))))
Af * (c * U) is non empty V16() V19( the carrier of ((TOP-REAL 2) | F)) V20( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))))
K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((TOP-REAL 2) | F), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)))) is non empty set
U1 is V16() Function-like FinSequence-like Element of the carrier of ((()) | F1)
(() | ((()) | F1)) . U1 is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),(3 / 2).[))
(Af * (c * U)) . U1 is complex real ext-real set
d is V16() Function-like FinSequence-like Element of the carrier of (())
() . d is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),(3 / 2).[))
n is complex real ext-real set
d1 is complex real ext-real set
|[n,d1]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos n is complex real ext-real Element of REAL
(arccos n) / (2 * PI) is complex real ext-real Element of REAL
1 + ((arccos n) / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos n) / (2 * PI)) is complex real ext-real Element of REAL
|[n,d1]| `1 is complex real ext-real Element of REAL
arccos . n is complex real ext-real set
c . n is complex real ext-real set
rng c is V67() V68() V69() Element of K6( the carrier of (R^1 | (E)))
K6( the carrier of (R^1 | (E))) is non empty set
(arccos . n) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos . n) / (2 * PI)) is complex real ext-real Element of REAL
(1 / (2 * PI)) * (arccos . n) is complex real ext-real Element of REAL
1 - ((1 / (2 * PI)) * (arccos . n)) is complex real ext-real Element of REAL
(- (1 / (2 * PI))) * (arccos . n) is complex real ext-real Element of REAL
((- (1 / (2 * PI))) * (arccos . n)) + 1 is complex real ext-real Element of REAL
(AffineMap ((- (1 / (2 * PI))),1)) . (arccos . n) is complex real ext-real set
Af . (arccos . n) is complex real ext-real set
Af . (c . n) is complex real ext-real set
c . (|[n,d1]| `1) is complex real ext-real set
Af . (c . (|[n,d1]| `1)) is complex real ext-real set
p . U1 is complex real ext-real set
c . (p . U1) is complex real ext-real set
Af . (c . (p . U1)) is complex real ext-real set
U . U1 is complex real ext-real set
c . (U . U1) is complex real ext-real set
Af . (c . (U . U1)) is complex real ext-real set
(c * U) . U1 is complex real ext-real set
Af . ((c * U) . U1) is complex real ext-real set
f is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
f `2 is complex real ext-real Element of REAL
F2 is set
F is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
F `2 is complex real ext-real Element of REAL
F2 is set
F is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
F `2 is complex real ext-real Element of REAL
D2 is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
D2 `2 is complex real ext-real Element of REAL
F2 is non empty Element of K6( the carrier of (()))
(()) | F2 is non empty strict TopSpace-like V221() V222() SubSpace of (())
U is V16() Function-like FinSequence-like Element of the carrier of (())
the carrier of ((()) | F2) is non empty set
F is non empty Element of K6( the carrier of (()))
(()) | F is non empty strict TopSpace-like V221() V222() SubSpace of (())
the carrier of ((()) | F) is non empty set
F2 \/ F is non empty Element of K6( the carrier of (()))
d1 is set
f is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
f `2 is complex real ext-real Element of REAL
() | ((()) | F) is non empty V16() V19( the carrier of ((()) | F)) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | F), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
K7( the carrier of ((()) | F), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((()) | F), the carrier of (R^1 | (].(1 / 2),(3 / 2).[)))) is non empty set
() | the carrier of ((()) | F) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
n is V16() Function-like FinSequence-like Element of the carrier of ((()) | F)
() | ((()) | F2) is non empty V16() V19( the carrier of ((()) | F2)) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of ((()) | F2), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
K7( the carrier of ((()) | F2), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of ((()) | F2), the carrier of (R^1 | (].(1 / 2),(3 / 2).[)))) is non empty set
() | the carrier of ((()) | F2) is V16() V19( the carrier of (())) V20( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) Function-like V51() V52() V53() Element of K6(K7( the carrier of (()), the carrier of (R^1 | (].(1 / 2),(3 / 2).[))))
U1 is V16() Function-like FinSequence-like Element of the carrier of ((()) | F2)
((()) | F2) union ((()) | F) is non empty strict TopSpace-like V221() V222() SubSpace of (())
(1 / (2 * PI)) (#) arccos is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
[.p0,0.[ is non empty V67() V68() V69() Element of K6(REAL)
].0,p1.] is non empty V67() V68() V69() Element of K6(REAL)
F is V16() Function-like FinSequence-like Element of the carrier of (())
[#] (Closed-Interval-TSpace ((- 1),1)) is non empty non proper V67() V68() V69() open closed dense non boundary Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
K6( the carrier of (Closed-Interval-TSpace ((- 1),1))) is non empty set
() . F is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),(3 / 2).[))
U1 is complex real ext-real set
d is complex real ext-real set
|[U1,d]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos U1 is complex real ext-real Element of REAL
(arccos U1) / (2 * PI) is complex real ext-real Element of REAL
1 + ((arccos U1) / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos U1) / (2 * PI)) is complex real ext-real Element of REAL
U is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
U `2 is complex real ext-real Element of REAL
U `1 is complex real ext-real Element of REAL
dom ((1 / (2 * PI)) (#) arccos) is V67() V68() V69() Element of K6(REAL)
((1 / (2 * PI)) (#) arccos) . U1 is complex real ext-real set
arccos . U1 is complex real ext-real set
(arccos . U1) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 * (arccos . U1) is complex real ext-real Element of REAL
(1 * (arccos . U1)) / (2 * PI) is complex real ext-real Element of REAL
K6( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) is non empty set
h1 is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
h1 - ((1 / (2 * PI)) (#) arccos) is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
d1 is V67() V68() V69() Element of K6( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)))
f is V67() V68() V69() Element of K6(REAL)
f1 is V67() V68() V69() Element of K6( the carrier of R^1)
J is open V67() V68() V69() Element of K6(REAL)
d2 is open V67() V68() V69() Neighbourhood of () . F
(h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.] is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . U1 is complex real ext-real set
(h1 - ((1 / (2 * PI)) (#) arccos)) . U1 is complex real ext-real set
dom (h1 - ((1 / (2 * PI)) (#) arccos)) is V67() V68() V69() Element of K6(REAL)
dom h1 is V67() V68() V69() Element of K6(REAL)
(dom h1) /\ (dom ((1 / (2 * PI)) (#) arccos)) is V67() V68() V69() Element of K6(REAL)
REAL /\ (dom ((1 / (2 * PI)) (#) arccos)) is V67() V68() V69() Element of K6(REAL)
dom ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) is V67() V68() V69() Element of K6(REAL)
(arccos . U1) / (2 * PI) is complex real ext-real Element of REAL
1 - ((arccos . U1) / (2 * PI)) is complex real ext-real Element of REAL
(1,2) --> (U1,d) is set
h1 . U1 is complex real ext-real set
(h1 . U1) - (((1 / (2 * PI)) (#) arccos) . U1) is complex real ext-real set
1 - ((1 * (arccos . U1)) / (2 * PI)) is complex real ext-real Element of REAL
F is open V67() V68() V69() Neighbourhood of U1
((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: F is V67() V68() V69() Element of K6(REAL)
F /\ [.(- 1),1.] is V67() V68() V69() Element of K6(REAL)
a is V67() V68() V69() Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
I is V67() V68() V69() Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
(1,2) --> (a,I) is non empty V16() V19({1,2}) V20(K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) Function-like total quasi_total Element of K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))))
{1,2} is non empty V67() V68() V69() V70() V71() V72() set
K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) is non empty set
K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1))))) is non empty set
product ((1,2) --> (a,I)) is set
(product ((1,2) --> (a,I))) /\ the carrier of (()) is set
W is Element of K6( the carrier of (()))
() .: W is V67() V68() V69() Element of K6( the carrier of (R^1 | (].(1 / 2),(3 / 2).[)))
K6( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) is non empty set
the carrier of (Trectangle (p0,p1,p0,p1)) is non empty set
K6( the carrier of (Trectangle (p0,p1,p0,p1))) is non empty set
((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: a is V67() V68() V69() Element of K6(REAL)
(F) is V67() V68() V69() open Element of K6( the carrier of R^1)
KK is Element of K6( the carrier of (Trectangle (p0,p1,p0,p1)))
M3 is V67() V68() V69() open Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
I3 is V67() V68() V69() open Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
(1,2) --> (M3,I3) is non empty V16() V19({1,2}) V20(K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) Function-like total quasi_total Element of K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))))
product ((1,2) --> (M3,I3)) is set
m is set
c is V16() Function-like FinSequence-like Element of the carrier of (())
() . c is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),(3 / 2).[))
c . 1 is set
c1 is complex real ext-real set
c2 is complex real ext-real set
|[c1,c2]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos c1 is complex real ext-real Element of REAL
(arccos c1) / (2 * PI) is complex real ext-real Element of REAL
1 + ((arccos c1) / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos c1) / (2 * PI)) is complex real ext-real Element of REAL
c . 2 is set
1 * (arccos c1) is complex real ext-real Element of REAL
(1 * (arccos c1)) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 - ((1 * (arccos c1)) * (1 / (2 * PI))) is complex real ext-real Element of REAL
((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . (c . 1) is complex real ext-real set
(h1 - ((1 / (2 * PI)) (#) arccos)) . (c . 1) is complex real ext-real set
h1 . (c . 1) is complex real ext-real set
((1 / (2 * PI)) (#) arccos) . (c . 1) is complex real ext-real set
(h1 . (c . 1)) - (((1 / (2 * PI)) (#) arccos) . (c . 1)) is complex real ext-real set
1 - (((1 / (2 * PI)) (#) arccos) . (c . 1)) is complex real ext-real Element of REAL
arccos . (c . 1) is complex real ext-real set
(arccos . (c . 1)) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos . (c . 1)) * (1 / (2 * PI))) is complex real ext-real Element of REAL
arccos . c1 is complex real ext-real set
(arccos . c1) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos . c1) * (1 / (2 * PI))) is complex real ext-real Element of REAL
(arccos c1) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos c1) * (1 / (2 * PI))) is complex real ext-real Element of REAL
K6( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) is non empty set
h1 is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
h1 + ((1 / (2 * PI)) (#) arccos) is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
d1 is V67() V68() V69() Element of K6( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)))
f is V67() V68() V69() Element of K6(REAL)
f1 is V67() V68() V69() Element of K6( the carrier of R^1)
J is open V67() V68() V69() Element of K6(REAL)
d2 is open V67() V68() V69() Neighbourhood of () . F
(h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.] is V16() V19( REAL ) V20( REAL ) Function-like V51() V52() V53() Element of K6(K7(REAL,REAL))
((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . U1 is complex real ext-real set
(h1 + ((1 / (2 * PI)) (#) arccos)) . U1 is complex real ext-real set
dom (h1 + ((1 / (2 * PI)) (#) arccos)) is V67() V68() V69() Element of K6(REAL)
dom h1 is V67() V68() V69() Element of K6(REAL)
(dom h1) /\ (dom ((1 / (2 * PI)) (#) arccos)) is V67() V68() V69() Element of K6(REAL)
REAL /\ (dom ((1 / (2 * PI)) (#) arccos)) is V67() V68() V69() Element of K6(REAL)
dom ((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) is V67() V68() V69() Element of K6(REAL)
(arccos . U1) / (2 * PI) is complex real ext-real Element of REAL
1 + ((arccos . U1) / (2 * PI)) is complex real ext-real Element of REAL
(1,2) --> (U1,d) is set
h1 . U1 is complex real ext-real set
(h1 . U1) + (((1 / (2 * PI)) (#) arccos) . U1) is complex real ext-real set
1 + ((1 * (arccos . U1)) / (2 * PI)) is complex real ext-real Element of REAL
F is open V67() V68() V69() Neighbourhood of U1
((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: F is V67() V68() V69() Element of K6(REAL)
F /\ [.(- 1),1.] is V67() V68() V69() Element of K6(REAL)
a is V67() V68() V69() Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
I is V67() V68() V69() Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
(1,2) --> (a,I) is non empty V16() V19({1,2}) V20(K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) Function-like total quasi_total Element of K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))))
{1,2} is non empty V67() V68() V69() V70() V71() V72() set
K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) is non empty set
K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1))))) is non empty set
product ((1,2) --> (a,I)) is set
(product ((1,2) --> (a,I))) /\ the carrier of (()) is set
W is Element of K6( the carrier of (()))
() .: W is V67() V68() V69() Element of K6( the carrier of (R^1 | (].(1 / 2),(3 / 2).[)))
K6( the carrier of (R^1 | (].(1 / 2),(3 / 2).[))) is non empty set
the carrier of (Trectangle (p0,p1,p0,p1)) is non empty set
K6( the carrier of (Trectangle (p0,p1,p0,p1))) is non empty set
((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: a is V67() V68() V69() Element of K6(REAL)
(F) is V67() V68() V69() open Element of K6( the carrier of R^1)
KK is Element of K6( the carrier of (Trectangle (p0,p1,p0,p1)))
M3 is V67() V68() V69() open Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
I3 is V67() V68() V69() open Element of K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))
(1,2) --> (M3,I3) is non empty V16() V19({1,2}) V20(K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))) Function-like total quasi_total Element of K6(K7({1,2},K6( the carrier of (Closed-Interval-TSpace ((- 1),1)))))
product ((1,2) --> (M3,I3)) is set
m is set
c is V16() Function-like FinSequence-like Element of the carrier of (())
() . c is complex real ext-real Element of the carrier of (R^1 | (].(1 / 2),(3 / 2).[))
c . 1 is set
c1 is complex real ext-real set
c2 is complex real ext-real set
|[c1,c2]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
arccos c1 is complex real ext-real Element of REAL
(arccos c1) / (2 * PI) is complex real ext-real Element of REAL
1 + ((arccos c1) / (2 * PI)) is complex real ext-real Element of REAL
1 - ((arccos c1) / (2 * PI)) is complex real ext-real Element of REAL
c . 2 is set
1 * (arccos c1) is complex real ext-real Element of REAL
(1 * (arccos c1)) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 + ((1 * (arccos c1)) * (1 / (2 * PI))) is complex real ext-real Element of REAL
((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . (c . 1) is complex real ext-real set
(h1 + ((1 / (2 * PI)) (#) arccos)) . (c . 1) is complex real ext-real set
h1 . (c . 1) is complex real ext-real set
((1 / (2 * PI)) (#) arccos) . (c . 1) is complex real ext-real set
(h1 . (c . 1)) + (((1 / (2 * PI)) (#) arccos) . (c . 1)) is complex real ext-real set
1 + (((1 / (2 * PI)) (#) arccos) . (c . 1)) is complex real ext-real Element of REAL
arccos . (c . 1) is complex real ext-real set
(arccos . (c . 1)) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 + ((arccos . (c . 1)) * (1 / (2 * PI))) is complex real ext-real Element of REAL
arccos . c1 is complex real ext-real set
(arccos . c1) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 + ((arccos . c1) * (1 / (2 * PI))) is complex real ext-real Element of REAL
(arccos c1) * (1 / (2 * PI)) is complex real ext-real Element of REAL
1 + ((arccos c1) * (1 / (2 * PI))) is complex real ext-real Element of REAL
D2 is complex real ext-real integer set
(D2) is complex real ext-real Element of the carrier of R^1
(D2) + 1 is complex real ext-real Element of REAL
].(D2),((D2) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].(D2),((D2) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(D2),((D2) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
() . (D2) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . (D2))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
((D2)) is non empty V16() V19( the carrier of (R^1 | (].(D2),((D2) + 1).[))) V20( the carrier of ((() . (D2)))) Function-like one-to-one total quasi_total onto bijective continuous Element of K6(K7( the carrier of (R^1 | (].(D2),((D2) + 1).[)), the carrier of ((() . (D2)))))
the carrier of (R^1 | (].(D2),((D2) + 1).[)) is non empty V67() V68() V69() set
the carrier of ((() . (D2))) is non empty set
K7( the carrier of (R^1 | (].(D2),((D2) + 1).[)), the carrier of ((() . (D2)))) is non empty set
K6(K7( the carrier of (R^1 | (].(D2),((D2) + 1).[)), the carrier of ((() . (D2))))) is non empty set
() | ].(D2),((D2) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
- D2 is complex real ext-real integer set
AffineMap (1,(- D2)) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
0 + D2 is complex real ext-real integer Element of REAL
(0 + D2) + p1 is complex real ext-real Element of REAL
].(0 + D2),((0 + D2) + p1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(AffineMap (1,(- D2))) | ].(0 + D2),((0 + D2) + p1).[ is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (AffineMap (1,(- D2))) is V67() V68() V69() Element of K6(REAL)
dom ((AffineMap (1,(- D2))) | ].(0 + D2),((0 + D2) + p1).[) is V67() V68() V69() Element of K6(REAL)
D2 + 1 is complex real ext-real integer Element of REAL
].D2,(D2 + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
rng ((AffineMap (1,(- D2))) | ].(0 + D2),((0 + D2) + p1).[) is V67() V68() V69() Element of K6(REAL)
0 + 1 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
].0,(0 + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].D2,(D2 + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].D2,(D2 + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].D2,(D2 + 1).[)) is non empty V67() V68() V69() set
D2 + p1 is complex real ext-real set
].D2,(D2 + p1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].D2,(D2 + p1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].D2,(D2 + p1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].D2,(D2 + p1).[)) is non empty V67() V68() V69() set
K7( the carrier of (R^1 | (].D2,(D2 + p1).[)), the carrier of (R^1 | (].0,(0 + p1).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | (].D2,(D2 + p1).[)), the carrier of (R^1 | (].0,(0 + p1).[)))) is non empty set
((0 + D2)) is complex real ext-real Element of the carrier of R^1
(((0 + D2))) is non empty V16() V19( the carrier of (R^1 | (].((0 + D2)),(((0 + D2)) + 1).[))) V20( the carrier of ((() . ((0 + D2))))) Function-like one-to-one total quasi_total onto bijective continuous Element of K6(K7( the carrier of (R^1 | (].((0 + D2)),(((0 + D2)) + 1).[)), the carrier of ((() . ((0 + D2))))))
((0 + D2)) + 1 is complex real ext-real Element of REAL
].((0 + D2)),(((0 + D2)) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].((0 + D2)),(((0 + D2)) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].((0 + D2)),(((0 + D2)) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].((0 + D2)),(((0 + D2)) + 1).[)) is non empty V67() V68() V69() set
() . ((0 + D2)) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . ((0 + D2)))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
the carrier of ((() . ((0 + D2)))) is non empty set
K7( the carrier of (R^1 | (].((0 + D2)),(((0 + D2)) + 1).[)), the carrier of ((() . ((0 + D2))))) is non empty set
K6(K7( the carrier of (R^1 | (].((0 + D2)),(((0 + D2)) + 1).[)), the carrier of ((() . ((0 + D2)))))) is non empty set
() | ].((0 + D2)),(((0 + D2)) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
F2 is non empty V16() V19( the carrier of (R^1 | (].D2,(D2 + p1).[))) V20( the carrier of (R^1 | (].0,(0 + p1).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | (].D2,(D2 + p1).[)), the carrier of (R^1 | (].0,(0 + p1).[))))
((0)) * F2 is V16() V19( the carrier of (R^1 | (].D2,(D2 + p1).[))) V20( the carrier of ((() . (0)))) Function-like Element of K6(K7( the carrier of (R^1 | (].D2,(D2 + p1).[)), the carrier of ((() . (0)))))
K7( the carrier of (R^1 | (].D2,(D2 + p1).[)), the carrier of ((() . (0)))) is non empty set
K6(K7( the carrier of (R^1 | (].D2,(D2 + p1).[)), the carrier of ((() . (0))))) is non empty set
rng (AffineMap (1,(- D2))) is V67() V68() V69() Element of K6(REAL)
((rng (AffineMap (1,(- D2))))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng (AffineMap (1,(- D2))))) is strict TopSpace-like V299() SubSpace of R^1
((dom (AffineMap (1,(- D2))))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom (AffineMap (1,(- D2))))) is strict TopSpace-like V299() SubSpace of R^1
((AffineMap (1,(- D2)))) is V16() V19( the carrier of (R^1 | ((dom (AffineMap (1,(- D2))))))) V19( the carrier of (R^1 | ((dom (AffineMap (1,(- D2))))))) V20( the carrier of (R^1 | ((rng (AffineMap (1,(- D2))))))) V20( the carrier of (R^1 | ((rng (AffineMap (1,(- D2))))))) Function-like one-to-one quasi_total quasi_total onto onto bijective bijective V51() V52() V53() continuous continuous open Element of K6(K7( the carrier of (R^1 | ((dom (AffineMap (1,(- D2)))))), the carrier of (R^1 | ((rng (AffineMap (1,(- D2))))))))
the carrier of (R^1 | ((dom (AffineMap (1,(- D2)))))) is V67() V68() V69() set
the carrier of (R^1 | ((rng (AffineMap (1,(- D2)))))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom (AffineMap (1,(- D2)))))), the carrier of (R^1 | ((rng (AffineMap (1,(- D2))))))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom (AffineMap (1,(- D2)))))), the carrier of (R^1 | ((rng (AffineMap (1,(- D2)))))))) is non empty set
(1 / 2) + D2 is complex real ext-real Element of REAL
(((1 / 2) + D2)) is complex real ext-real Element of the carrier of R^1
(((1 / 2) + D2)) + 1 is complex real ext-real Element of REAL
].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
() . (((1 / 2) + D2)) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . (((1 / 2) + D2)))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
((((1 / 2) + D2))) is non empty V16() V19( the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[))) V20( the carrier of ((() . (((1 / 2) + D2))))) Function-like one-to-one total quasi_total onto bijective continuous Element of K6(K7( the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[)), the carrier of ((() . (((1 / 2) + D2))))))
the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[)) is non empty V67() V68() V69() set
the carrier of ((() . (((1 / 2) + D2)))) is non empty set
K7( the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[)), the carrier of ((() . (((1 / 2) + D2))))) is non empty set
K6(K7( the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[)), the carrier of ((() . (((1 / 2) + D2)))))) is non empty set
() | ].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
(1 / 2) - 1 is complex real ext-real Element of REAL
[\(1 / 2)/] is complex real ext-real integer set
(1 / 2) - [\(1 / 2)/] is complex real ext-real Element of REAL
frac ((1 / 2) + D2) is complex real ext-real Element of REAL
frac (1 / 2) is complex real ext-real Element of REAL
(1 / 2) + 0 is non empty complex real ext-real positive non negative Element of REAL
(((1 / 2) + 0)) is complex real ext-real Element of the carrier of R^1
() . (((1 / 2) + 0)) is V16() Function-like FinSequence-like Element of the carrier of (2)
- D2 is complex real ext-real integer set
AffineMap (1,(- D2)) is non empty V16() V19( REAL ) V20( REAL ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
((1 / 2) + D2) + p1 is complex real ext-real Element of REAL
].((1 / 2) + D2),(((1 / 2) + D2) + p1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(AffineMap (1,(- D2))) | ].((1 / 2) + D2),(((1 / 2) + D2) + p1).[ is V16() V19( REAL ) V20( REAL ) Function-like continuous V51() V52() V53() Element of K6(K7(REAL,REAL))
dom (AffineMap (1,(- D2))) is V67() V68() V69() Element of K6(REAL)
dom ((AffineMap (1,(- D2))) | ].((1 / 2) + D2),(((1 / 2) + D2) + p1).[) is V67() V68() V69() Element of K6(REAL)
((1 / 2) + D2) + 1 is complex real ext-real Element of REAL
].((1 / 2) + D2),(((1 / 2) + D2) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
rng ((AffineMap (1,(- D2))) | ].((1 / 2) + D2),(((1 / 2) + D2) + p1).[) is V67() V68() V69() Element of K6(REAL)
(1 / 2) + 1 is non empty complex real ext-real positive non negative Element of REAL
].(1 / 2),((1 / 2) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].((1 / 2) + D2),(((1 / 2) + D2) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + 1).[)) is non empty V67() V68() V69() set
(].((1 / 2) + D2),(((1 / 2) + D2) + p1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + p1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + p1).[)) is non empty V67() V68() V69() set
K7( the carrier of (R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + p1).[)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) is non empty V51() V52() V53() set
K6(K7( the carrier of (R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + p1).[)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[)))) is non empty set
F2 is non empty V16() V19( the carrier of (R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + p1).[))) V20( the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))) Function-like total quasi_total V51() V52() V53() Element of K6(K7( the carrier of (R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + p1).[)), the carrier of (R^1 | (].(1 / 2),((1 / 2) + p1).[))))
(((1 / 2))) * F2 is V16() V19( the carrier of (R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + p1).[))) V20( the carrier of ((() . ((1 / 2))))) Function-like Element of K6(K7( the carrier of (R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + p1).[)), the carrier of ((() . ((1 / 2))))))
K7( the carrier of (R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + p1).[)), the carrier of ((() . ((1 / 2))))) is non empty set
K6(K7( the carrier of (R^1 | (].((1 / 2) + D2),(((1 / 2) + D2) + p1).[)), the carrier of ((() . ((1 / 2)))))) is non empty set
rng (AffineMap (1,(- D2))) is V67() V68() V69() Element of K6(REAL)
((rng (AffineMap (1,(- D2))))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((rng (AffineMap (1,(- D2))))) is strict TopSpace-like V299() SubSpace of R^1
((dom (AffineMap (1,(- D2))))) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | ((dom (AffineMap (1,(- D2))))) is strict TopSpace-like V299() SubSpace of R^1
((AffineMap (1,(- D2)))) is V16() V19( the carrier of (R^1 | ((dom (AffineMap (1,(- D2))))))) V19( the carrier of (R^1 | ((dom (AffineMap (1,(- D2))))))) V20( the carrier of (R^1 | ((rng (AffineMap (1,(- D2))))))) V20( the carrier of (R^1 | ((rng (AffineMap (1,(- D2))))))) Function-like one-to-one quasi_total quasi_total onto onto bijective bijective V51() V52() V53() continuous continuous open Element of K6(K7( the carrier of (R^1 | ((dom (AffineMap (1,(- D2)))))), the carrier of (R^1 | ((rng (AffineMap (1,(- D2))))))))
the carrier of (R^1 | ((dom (AffineMap (1,(- D2)))))) is V67() V68() V69() set
the carrier of (R^1 | ((rng (AffineMap (1,(- D2)))))) is V67() V68() V69() set
K7( the carrier of (R^1 | ((dom (AffineMap (1,(- D2)))))), the carrier of (R^1 | ((rng (AffineMap (1,(- D2))))))) is V51() V52() V53() set
K6(K7( the carrier of (R^1 | ((dom (AffineMap (1,(- D2)))))), the carrier of (R^1 | ((rng (AffineMap (1,(- D2)))))))) is non empty set
D2 is complex real ext-real integer set
(1 / 2) + D2 is complex real ext-real Element of REAL
(((1 / 2) + D2)) is complex real ext-real Element of the carrier of R^1
(((1 / 2) + D2)) + 1 is complex real ext-real Element of REAL
].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
() . (((1 / 2) + D2)) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . (((1 / 2) + D2)))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
((((1 / 2) + D2))) is non empty V16() V19( the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[))) V19( the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[))) V20( the carrier of ((() . (((1 / 2) + D2))))) V20( the carrier of ((() . (((1 / 2) + D2))))) Function-like one-to-one total total quasi_total quasi_total onto onto bijective bijective continuous continuous open Element of K6(K7( the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[)), the carrier of ((() . (((1 / 2) + D2))))))
the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[)) is non empty V67() V68() V69() set
the carrier of ((() . (((1 / 2) + D2)))) is non empty set
K7( the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[)), the carrier of ((() . (((1 / 2) + D2))))) is non empty set
K6(K7( the carrier of (R^1 | (].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[)), the carrier of ((() . (((1 / 2) + D2)))))) is non empty set
() | ].(((1 / 2) + D2)),((((1 / 2) + D2)) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
K6(K6( the carrier of (2))) is non empty set
() .: ].0,1.[ is Element of K6( the carrier of (2))
() .: ].(1 / 2),(3 / 2).[ is Element of K6( the carrier of (2))
{(() .: ].0,1.[),(() .: ].(1 / 2),(3 / 2).[)} is non empty Element of K6(K6( the carrier of (2)))
(0,1) is non empty open Element of K6(K6( the carrier of R^1))
{ ].(0 + b₁),(1 + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
union (0,1) is V67() V68() V69() Element of K6( the carrier of R^1)
((1 / 2),(3 / 2)) is non empty open Element of K6(K6( the carrier of R^1))
{ ].((1 / 2) + b₁),((3 / 2) + b₁).[ where b₁ is complex real ext-real integer V66() Element of INT : verum } is set
union ((1 / 2),(3 / 2)) is V67() V68() V69() Element of K6( the carrier of R^1)
() .: (union (0,1)) is Element of K6( the carrier of (2))
() .: (union ((1 / 2),(3 / 2))) is Element of K6( the carrier of (2))
{(() .: (union (0,1))),(() .: (union ((1 / 2),(3 / 2))))} is non empty Element of K6(K6( the carrier of (2)))
F is Element of K6(K6( the carrier of (2)))
(1 / 2) + 0 is non empty complex real ext-real positive non negative Element of REAL
(3 / 2) + 0 is non empty complex real ext-real positive non negative Element of REAL
].((1 / 2) + 0),((3 / 2) + 0).[ is open V67() V68() V69() Element of K6(REAL)
0 + 0 is empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V67() V68() V69() V70() V71() V72() V73() Element of REAL
1 + 0 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
].(0 + 0),(1 + 0).[ is non empty open V67() V68() V69() Element of K6(REAL)
0 + 1 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
[.0,(0 + 1).[ is non empty V67() V68() V69() Element of K6(REAL)
U is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | U is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | U) is V67() V68() V69() set
K7( the carrier of (R^1 | U), the carrier of (2)) is set
K6(K7( the carrier of (R^1 | U), the carrier of (2))) is non empty set
() | U is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
d is set
union F is set
U1 is V16() V19( the carrier of (R^1 | U)) V20( the carrier of (2)) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | U), the carrier of (2)))
rng U1 is Element of K6( the carrier of (2))
dom U1 is V67() V68() V69() Element of K6( the carrier of (R^1 | U))
K6( the carrier of (R^1 | U)) is non empty set
n is set
U1 . n is set
d1 is complex real ext-real Element of REAL
() . d1 is set
U1 . d1 is set
U1 . 0 is set
() . 0 is set
() . 1 is set
union F is Element of K6( the carrier of (2))
union F is Element of K6( the carrier of (2))
union F is Element of K6( the carrier of (2))
U1 is V67() V68() V69() Element of K6(REAL)
(U1) is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | (U1) is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | (U1)) is V67() V68() V69() set
K6( the carrier of (R^1 | (U1))) is non empty set
U is complex real ext-real integer set
U + 1 is complex real ext-real integer Element of REAL
].U,(U + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].U,(U + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].U,(U + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].U,(U + 1).[)) is non empty V67() V68() V69() set
K6( the carrier of (R^1 | (].U,(U + 1).[))) is non empty set
(1 / 2) + U is complex real ext-real Element of REAL
((1 / 2) + U) + 1 is complex real ext-real Element of REAL
].((1 / 2) + U),(((1 / 2) + U) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].((1 / 2) + U),(((1 / 2) + U) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].((1 / 2) + U),(((1 / 2) + U) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].((1 / 2) + U),(((1 / 2) + U) + 1).[)) is non empty V67() V68() V69() set
K6( the carrier of (R^1 | (].((1 / 2) + U),(((1 / 2) + U) + 1).[))) is non empty set
n is Element of K6( the carrier of (2))
d1 is open V67() V68() V69() Element of K6(REAL)
(d1) is V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (d1) is strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (d1)) is V67() V68() V69() set
K6( the carrier of (R^1 | (d1))) is non empty set
f is V67() V68() V69() Element of K6( the carrier of (R^1 | (d1)))
[#] (R^1 | (d1)) is non proper V67() V68() V69() open closed dense Element of K6( the carrier of (R^1 | (d1)))
([#] (R^1 | (d1))) /\ (d1) is V67() V68() V69() Element of K6( the carrier of R^1)
U1 is V67() V68() V69() Element of K6( the carrier of (R^1 | (].U,(U + 1).[)))
(U) is complex real ext-real Element of the carrier of R^1
() . (U) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . (U))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
(U) + 1 is complex real ext-real Element of REAL
].(U),((U) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].(U),((U) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(U),((U) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].(U),((U) + 1).[)) is non empty V67() V68() V69() set
the carrier of ((() . (U))) is non empty set
((U)) is non empty V16() V19( the carrier of (R^1 | (].(U),((U) + 1).[))) V19( the carrier of (R^1 | (].(U),((U) + 1).[))) V20( the carrier of ((() . (U)))) V20( the carrier of ((() . (U)))) Function-like one-to-one total total quasi_total quasi_total onto onto bijective bijective continuous continuous open Element of K6(K7( the carrier of (R^1 | (].(U),((U) + 1).[)), the carrier of ((() . (U)))))
K7( the carrier of (R^1 | (].(U),((U) + 1).[)), the carrier of ((() . (U)))) is non empty set
K6(K7( the carrier of (R^1 | (].(U),((U) + 1).[)), the carrier of ((() . (U))))) is non empty set
() | ].(U),((U) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
((U)) .: U1 is Element of K6( the carrier of ((() . (U))))
K6( the carrier of ((() . (U)))) is non empty set
d is V67() V68() V69() Element of K6( the carrier of (R^1 | (].((1 / 2) + U),(((1 / 2) + U) + 1).[)))
(((1 / 2) + U)) is complex real ext-real Element of the carrier of R^1
() . (((1 / 2) + U)) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . (((1 / 2) + U)))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
(((1 / 2) + U)) + 1 is complex real ext-real Element of REAL
].(((1 / 2) + U)),((((1 / 2) + U)) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].(((1 / 2) + U)),((((1 / 2) + U)) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(((1 / 2) + U)),((((1 / 2) + U)) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
the carrier of (R^1 | (].(((1 / 2) + U)),((((1 / 2) + U)) + 1).[)) is non empty V67() V68() V69() set
the carrier of ((() . (((1 / 2) + U)))) is non empty set
((((1 / 2) + U))) is non empty V16() V19( the carrier of (R^1 | (].(((1 / 2) + U)),((((1 / 2) + U)) + 1).[))) V19( the carrier of (R^1 | (].(((1 / 2) + U)),((((1 / 2) + U)) + 1).[))) V20( the carrier of ((() . (((1 / 2) + U))))) V20( the carrier of ((() . (((1 / 2) + U))))) Function-like one-to-one total total quasi_total quasi_total onto onto bijective bijective continuous continuous open Element of K6(K7( the carrier of (R^1 | (].(((1 / 2) + U)),((((1 / 2) + U)) + 1).[)), the carrier of ((() . (((1 / 2) + U))))))
K7( the carrier of (R^1 | (].(((1 / 2) + U)),((((1 / 2) + U)) + 1).[)), the carrier of ((() . (((1 / 2) + U))))) is non empty set
K6(K7( the carrier of (R^1 | (].(((1 / 2) + U)),((((1 / 2) + U)) + 1).[)), the carrier of ((() . (((1 / 2) + U)))))) is non empty set
() | ].(((1 / 2) + U)),((((1 / 2) + U)) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
((((1 / 2) + U))) .: d is Element of K6( the carrier of ((() . (((1 / 2) + U)))))
K6( the carrier of ((() . (((1 / 2) + U))))) is non empty set
(((1 / 2))) .: ].(1 / 2),(3 / 2).[ is Element of K6( the carrier of ((() . ((1 / 2)))))
K6( the carrier of ((() . ((1 / 2))))) is non empty set
((0)) .: ].0,1.[ is Element of K6( the carrier of ((() . (0))))
K6( the carrier of ((() . (0)))) is non empty set
U is Element of K6( the carrier of (2))
() " U is V67() V68() V69() Element of K6( the carrier of R^1)
(2) | U is strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of ((2) | U) is set
[#] ((2) | U) is non proper open closed dense Element of K6( the carrier of ((2) | U))
K6( the carrier of ((2) | U)) is non empty set
n is complex real ext-real Element of the carrier of R^1
[\n/] is complex real ext-real integer set
0 + [\n/] is complex real ext-real integer Element of REAL
1 + [\n/] is complex real ext-real integer Element of REAL
].(0 + [\n/]),(1 + [\n/]).[ is open V67() V68() V69() Element of K6(REAL)
d is complex real ext-real Element of the carrier of R^1
f1 is set
J is complex real ext-real integer V66() Element of INT
0 + J is complex real ext-real integer Element of REAL
1 + J is complex real ext-real integer Element of REAL
].(0 + J),(1 + J).[ is open V67() V68() V69() Element of K6(REAL)
(1 + J) - 1 is complex real ext-real integer Element of REAL
d - 1 is complex real ext-real Element of REAL
J - J is complex real ext-real integer set
(d - 1) - J is complex real ext-real Element of REAL
0 + 1 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
d - J is complex real ext-real set
(d - J) - 1 is complex real ext-real Element of REAL
((d - J) - 1) + 1 is complex real ext-real Element of REAL
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * (d - J) is complex real ext-real Element of REAL
n - [\n/] is complex real ext-real set
() . (n - [\n/]) is set
(2 * PI) * (n - [\n/]) is complex real ext-real Element of REAL
cos ((2 * PI) * (n - [\n/])) is complex real ext-real Element of REAL
sin ((2 * PI) * (n - [\n/])) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (n - [\n/]))),(sin ((2 * PI) * (n - [\n/])))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
n - 1 is complex real ext-real Element of REAL
[\n/] - [\n/] is complex real ext-real integer set
(n - 1) - [\n/] is complex real ext-real Element of REAL
((n - 1) - [\n/]) + 1 is complex real ext-real Element of REAL
() . d is V16() Function-like FinSequence-like Element of the carrier of (2)
() . n is V16() Function-like FinSequence-like Element of the carrier of (2)
- [\n/] is complex real ext-real integer set
n + (- [\n/]) is complex real ext-real set
() . (n + (- [\n/])) is set
[\d/] is complex real ext-real integer set
() . (d - J) is set
cos ((2 * PI) * (d - J)) is complex real ext-real Element of REAL
sin ((2 * PI) * (d - J)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (d - J))),(sin ((2 * PI) * (d - J)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
- J is complex real ext-real integer set
d + (- J) is complex real ext-real set
() . (d + (- J)) is set
(d - J) + [\n/] is complex real ext-real set
d is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | d is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | d) is V67() V68() V69() set
K7( the carrier of (R^1 | d), the carrier of ((2) | U)) is set
K6(K7( the carrier of (R^1 | d), the carrier of ((2) | U))) is non empty set
() | d is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
n is complex real ext-real integer V66() Element of INT
0 + n is complex real ext-real integer Element of REAL
1 + n is complex real ext-real integer Element of REAL
].(0 + n),(1 + n).[ is open V67() V68() V69() Element of K6(REAL)
n + p1 is complex real ext-real set
].n,(n + p1).[ is non empty open V67() V68() V69() Element of K6(REAL)
f is non empty V67() V68() V69() Element of K6( the carrier of R^1)
() | f is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
d1 is non empty V67() V68() V69() Element of K6( the carrier of R^1)
(() | f) | d1 is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
f1 is V16() V19( the carrier of (R^1 | d)) V20( the carrier of ((2) | U)) Function-like quasi_total Element of K6(K7( the carrier of (R^1 | d), the carrier of ((2) | U)))
R^1 | d1 is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | d1) is non empty V67() V68() V69() set
U1 is non empty Element of K6( the carrier of (2))
(2) | U1 is non empty strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of ((2) | U1) is non empty set
K7( the carrier of (R^1 | d1), the carrier of ((2) | U1)) is non empty set
K6(K7( the carrier of (R^1 | d1), the carrier of ((2) | U1))) is non empty set
R^1 | f is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | f) is non empty V67() V68() V69() set
K6( the carrier of (R^1 | f)) is non empty set
d2 is V67() V68() V69() Element of K6( the carrier of (R^1 | f))
(R^1 | f) | d2 is strict TopSpace-like V299() SubSpace of R^1 | f
K7( the carrier of (R^1 | f), the carrier of (2)) is non empty set
K6(K7( the carrier of (R^1 | f), the carrier of (2))) is non empty set
(n) is complex real ext-real Element of the carrier of R^1
(n) + 1 is complex real ext-real Element of REAL
].(n),((n) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].(n),((n) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(n),((n) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
() . (n) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . (n))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
((n)) is non empty V16() V19( the carrier of (R^1 | (].(n),((n) + 1).[))) V19( the carrier of (R^1 | (].(n),((n) + 1).[))) V20( the carrier of ((() . (n)))) V20( the carrier of ((() . (n)))) Function-like one-to-one total total quasi_total quasi_total onto onto bijective bijective continuous continuous open Element of K6(K7( the carrier of (R^1 | (].(n),((n) + 1).[)), the carrier of ((() . (n)))))
the carrier of (R^1 | (].(n),((n) + 1).[)) is non empty V67() V68() V69() set
the carrier of ((() . (n))) is non empty set
K7( the carrier of (R^1 | (].(n),((n) + 1).[)), the carrier of ((() . (n)))) is non empty set
K6(K7( the carrier of (R^1 | (].(n),((n) + 1).[)), the carrier of ((() . (n))))) is non empty set
() | ].(n),((n) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
F is non empty V16() V19( the carrier of (R^1 | f)) V20( the carrier of (2)) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | f), the carrier of (2)))
() .: d is Element of K6( the carrier of (2))
J is non empty V16() V19( the carrier of (R^1 | d1)) V20( the carrier of ((2) | U1)) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | d1), the carrier of ((2) | U1)))
rng J is Element of K6( the carrier of ((2) | U1))
K6( the carrier of ((2) | U1)) is non empty set
a is set
dom (() | d) is V67() V68() V69() Element of K6( the carrier of R^1)
a is complex real ext-real Element of the carrier of R^1
() . a is V16() Function-like FinSequence-like Element of the carrier of (2)
(() | d) . a is set
n is complex real ext-real Element of the carrier of R^1
[\n/] is complex real ext-real integer set
d is complex real ext-real Element of the carrier of R^1
f is set
f1 is complex real ext-real integer V66() Element of INT
(1 / 2) + f1 is complex real ext-real Element of REAL
(3 / 2) + f1 is complex real ext-real Element of REAL
].((1 / 2) + f1),((3 / 2) + f1).[ is open V67() V68() V69() Element of K6(REAL)
0 + f1 is complex real ext-real integer Element of REAL
() . d is V16() Function-like FinSequence-like Element of the carrier of (2)
() . n is V16() Function-like FinSequence-like Element of the carrier of (2)
d - f1 is complex real ext-real set
1 + f1 is complex real ext-real integer Element of REAL
J is complex real ext-real integer V66() Element of INT
(1 / 2) + J is complex real ext-real Element of REAL
0 + J is complex real ext-real integer Element of REAL
J - 1 is complex real ext-real integer Element of REAL
(1 / 2) + (J - 1) is complex real ext-real Element of REAL
(3 / 2) + (J - 1) is complex real ext-real Element of REAL
].((1 / 2) + (J - 1)),((3 / 2) + (J - 1)).[ is open V67() V68() V69() Element of K6(REAL)
- (1 / 2) is non empty complex real ext-real non positive negative Element of REAL
(- (1 / 2)) + J is complex real ext-real Element of REAL
1 + f1 is complex real ext-real integer Element of REAL
f1 + 1 is complex real ext-real integer Element of REAL
(f1 + 1) - 1 is complex real ext-real integer Element of REAL
d - 1 is complex real ext-real Element of REAL
f1 - f1 is complex real ext-real integer set
(d - 1) - f1 is complex real ext-real Element of REAL
0 + 1 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
(d - f1) - 1 is complex real ext-real Element of REAL
((d - f1) - 1) + 1 is complex real ext-real Element of REAL
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * (d - f1) is complex real ext-real Element of REAL
(1 / 2) + [\n/] is complex real ext-real Element of REAL
(3 / 2) + [\n/] is complex real ext-real Element of REAL
].((1 / 2) + [\n/]),((3 / 2) + [\n/]).[ is open V67() V68() V69() Element of K6(REAL)
n - n is complex real ext-real set
n - [\n/] is complex real ext-real set
() . (n - [\n/]) is set
(2 * PI) * (n - [\n/]) is complex real ext-real Element of REAL
cos ((2 * PI) * (n - [\n/])) is complex real ext-real Element of REAL
sin ((2 * PI) * (n - [\n/])) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (n - [\n/]))),(sin ((2 * PI) * (n - [\n/])))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
n - 1 is complex real ext-real Element of REAL
[\n/] - [\n/] is complex real ext-real integer set
(n - 1) - [\n/] is complex real ext-real Element of REAL
((n - 1) - [\n/]) + 1 is complex real ext-real Element of REAL
- [\n/] is complex real ext-real integer set
n + (- [\n/]) is complex real ext-real set
() . (n + (- [\n/])) is set
((1 / 2) + f1) - f1 is complex real ext-real Element of REAL
(d - f1) + [\n/] is complex real ext-real set
() . (d - f1) is set
cos ((2 * PI) * (d - f1)) is complex real ext-real Element of REAL
sin ((2 * PI) * (d - f1)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (d - f1))),(sin ((2 * PI) * (d - f1)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
- f1 is complex real ext-real integer set
d + (- f1) is complex real ext-real set
() . (d + (- f1)) is set
1 + f1 is complex real ext-real integer Element of REAL
d - 1 is complex real ext-real Element of REAL
f1 + 1 is complex real ext-real integer Element of REAL
(f1 + 1) - 1 is complex real ext-real integer Element of REAL
(d - 1) - f1 is complex real ext-real Element of REAL
f1 - f1 is complex real ext-real integer set
[\n/] - 1 is complex real ext-real integer Element of REAL
(1 / 2) + ([\n/] - 1) is complex real ext-real Element of REAL
(3 / 2) + ([\n/] - 1) is complex real ext-real Element of REAL
].((1 / 2) + ([\n/] - 1)),((3 / 2) + ([\n/] - 1)).[ is open V67() V68() V69() Element of K6(REAL)
- (1 / 2) is non empty complex real ext-real non positive negative Element of REAL
((d - 1) - f1) + [\n/] is complex real ext-real Element of REAL
(- (1 / 2)) + [\n/] is complex real ext-real Element of REAL
(3 / 2) - 1 is complex real ext-real Element of REAL
(d - f1) - 1 is complex real ext-real Element of REAL
n - n is complex real ext-real set
n - [\n/] is complex real ext-real set
() . (n - [\n/]) is set
(2 * PI) * (n - [\n/]) is complex real ext-real Element of REAL
cos ((2 * PI) * (n - [\n/])) is complex real ext-real Element of REAL
sin ((2 * PI) * (n - [\n/])) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * (n - [\n/]))),(sin ((2 * PI) * (n - [\n/])))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
((3 / 2) + f1) - 1 is complex real ext-real Element of REAL
((1 / 2) + f1) - f1 is complex real ext-real Element of REAL
(2 * PI) * 1 is non empty complex real ext-real positive non negative Element of REAL
(2 * PI) * ((d - 1) - f1) is complex real ext-real Element of REAL
- [\n/] is complex real ext-real integer set
n + (- [\n/]) is complex real ext-real set
() . (n + (- [\n/])) is set
n - 1 is complex real ext-real Element of REAL
[\n/] - [\n/] is complex real ext-real integer set
(n - 1) - [\n/] is complex real ext-real Element of REAL
0 + 1 is non empty ordinal natural complex real ext-real positive non negative integer Element of REAL
((n - 1) - [\n/]) + 1 is complex real ext-real Element of REAL
() . ((d - 1) - f1) is set
cos ((2 * PI) * ((d - 1) - f1)) is complex real ext-real Element of REAL
sin ((2 * PI) * ((d - 1) - f1)) is complex real ext-real Element of REAL
|[(cos ((2 * PI) * ((d - 1) - f1))),(sin ((2 * PI) * ((d - 1) - f1)))]| is V16() Function-like V34(2) V51() V52() V53() FinSequence-like Element of the carrier of (TOP-REAL 2)
(- 1) - f1 is complex real ext-real integer Element of REAL
d + ((- 1) - f1) is complex real ext-real Element of REAL
() . (d + ((- 1) - f1)) is set
(1 / 2) + [\n/] is complex real ext-real Element of REAL
1 + f1 is complex real ext-real integer Element of REAL
d is V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | d is strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | d) is V67() V68() V69() set
K7( the carrier of (R^1 | d), the carrier of ((2) | U)) is set
K6(K7( the carrier of (R^1 | d), the carrier of ((2) | U))) is non empty set
() | d is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
n is complex real ext-real integer V66() Element of INT
(1 / 2) + n is complex real ext-real Element of REAL
(3 / 2) + n is complex real ext-real Element of REAL
].((1 / 2) + n),((3 / 2) + n).[ is open V67() V68() V69() Element of K6(REAL)
1 + n is complex real ext-real integer Element of REAL
[#] ((2) | U) is non proper open closed dense Element of K6( the carrier of ((2) | U))
K6( the carrier of ((2) | U)) is non empty set
f is V16() V19( the carrier of (R^1 | d)) V20( the carrier of ((2) | U)) Function-like quasi_total Element of K6(K7( the carrier of (R^1 | d), the carrier of ((2) | U)))
d1 is non empty V67() V68() V69() Element of K6( the carrier of R^1)
R^1 | d1 is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | d1) is non empty V67() V68() V69() set
U1 is non empty Element of K6( the carrier of (2))
(2) | U1 is non empty strict TopSpace-like V221() V222() SubSpace of (2)
the carrier of ((2) | U1) is non empty set
K7( the carrier of (R^1 | d1), the carrier of ((2) | U1)) is non empty set
K6(K7( the carrier of (R^1 | d1), the carrier of ((2) | U1))) is non empty set
((1 / 2) + n) + p1 is complex real ext-real Element of REAL
].((1 / 2) + n),(((1 / 2) + n) + p1).[ is non empty open V67() V68() V69() Element of K6(REAL)
J is non empty V67() V68() V69() Element of K6( the carrier of R^1)
() | J is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
(() | J) | d1 is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like Element of K6(K7( the carrier of R^1, the carrier of (2)))
R^1 | J is non empty strict TopSpace-like V299() SubSpace of R^1
the carrier of (R^1 | J) is non empty V67() V68() V69() set
K6( the carrier of (R^1 | J)) is non empty set
d2 is V67() V68() V69() Element of K6( the carrier of (R^1 | J))
(R^1 | J) | d2 is strict TopSpace-like V299() SubSpace of R^1 | J
() .: d is Element of K6( the carrier of (2))
f1 is non empty V16() V19( the carrier of (R^1 | d1)) V20( the carrier of ((2) | U1)) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | d1), the carrier of ((2) | U1)))
rng f1 is Element of K6( the carrier of ((2) | U1))
K6( the carrier of ((2) | U1)) is non empty set
F is set
dom (() | d) is V67() V68() V69() Element of K6( the carrier of R^1)
a is complex real ext-real Element of the carrier of R^1
() . a is V16() Function-like FinSequence-like Element of the carrier of (2)
(() | d) . a is set
K7( the carrier of (R^1 | J), the carrier of (2)) is non empty set
K6(K7( the carrier of (R^1 | J), the carrier of (2))) is non empty set
(((1 / 2) + n)) is complex real ext-real Element of the carrier of R^1
(((1 / 2) + n)) + 1 is complex real ext-real Element of REAL
].(((1 / 2) + n)),((((1 / 2) + n)) + 1).[ is non empty open V67() V68() V69() Element of K6(REAL)
(].(((1 / 2) + n)),((((1 / 2) + n)) + 1).[) is non empty V67() V68() V69() open Element of K6( the carrier of R^1)
R^1 | (].(((1 / 2) + n)),((((1 / 2) + n)) + 1).[) is non empty strict TopSpace-like open V299() SubSpace of R^1
() . (((1 / 2) + n)) is V16() Function-like FinSequence-like Element of the carrier of (2)
((() . (((1 / 2) + n)))) is non empty strict TopSpace-like open V221() V222() SubSpace of (2)
((((1 / 2) + n))) is non empty V16() V19( the carrier of (R^1 | (].(((1 / 2) + n)),((((1 / 2) + n)) + 1).[))) V19( the carrier of (R^1 | (].(((1 / 2) + n)),((((1 / 2) + n)) + 1).[))) V20( the carrier of ((() . (((1 / 2) + n))))) V20( the carrier of ((() . (((1 / 2) + n))))) Function-like one-to-one total total quasi_total quasi_total onto onto bijective bijective continuous continuous open Element of K6(K7( the carrier of (R^1 | (].(((1 / 2) + n)),((((1 / 2) + n)) + 1).[)), the carrier of ((() . (((1 / 2) + n))))))
the carrier of (R^1 | (].(((1 / 2) + n)),((((1 / 2) + n)) + 1).[)) is non empty V67() V68() V69() set
the carrier of ((() . (((1 / 2) + n)))) is non empty set
K7( the carrier of (R^1 | (].(((1 / 2) + n)),((((1 / 2) + n)) + 1).[)), the carrier of ((() . (((1 / 2) + n))))) is non empty set
K6(K7( the carrier of (R^1 | (].(((1 / 2) + n)),((((1 / 2) + n)) + 1).[)), the carrier of ((() . (((1 / 2) + n)))))) is non empty set
() | ].(((1 / 2) + n)),((((1 / 2) + n)) + 1).[ is V16() V19( the carrier of R^1) V20( the carrier of (2)) Function-like one-to-one Element of K6(K7( the carrier of R^1, the carrier of (2)))
F is non empty V16() V19( the carrier of (R^1 | J)) V20( the carrier of (2)) Function-like total quasi_total Element of K6(K7( the carrier of (R^1 | J), the carrier of (2)))