REAL is non empty V34() V158() V159() V160() V164() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() Element of K6(REAL)
K6(REAL) is set
COMPLEX is non empty V34() V158() V164() set
omega is non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() set
K6(omega) is set
K6(NAT) is set
RAT is non empty V34() V158() V159() V160() V161() V164() set
INT is non empty V34() V158() V159() V160() V161() V162() V164() set
K7(COMPLEX,COMPLEX) is V148() set
K6(K7(COMPLEX,COMPLEX)) is set
K7(K7(COMPLEX,COMPLEX),COMPLEX) is V148() set
K6(K7(K7(COMPLEX,COMPLEX),COMPLEX)) is set
K7(REAL,REAL) is V148() V149() V150() set
K6(K7(REAL,REAL)) is set
K7(K7(REAL,REAL),REAL) is V148() V149() V150() set
K6(K7(K7(REAL,REAL),REAL)) is set
K7(RAT,RAT) is V23( RAT ) V148() V149() V150() set
K6(K7(RAT,RAT)) is set
K7(K7(RAT,RAT),RAT) is V23( RAT ) V148() V149() V150() set
K6(K7(K7(RAT,RAT),RAT)) is set
K7(INT,INT) is V23( RAT ) V23( INT ) V148() V149() V150() set
K6(K7(INT,INT)) is set
K7(K7(INT,INT),INT) is V23( RAT ) V23( INT ) V148() V149() V150() set
K6(K7(K7(INT,INT),INT)) is set
K7(NAT,NAT) is V23( RAT ) V23( INT ) V148() V149() V150() V151() set
K7(K7(NAT,NAT),NAT) is V23( RAT ) V23( INT ) V148() V149() V150() V151() set
K6(K7(K7(NAT,NAT),NAT)) is set
K295() is set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
K455() is non empty V107() L9()
the carrier of K455() is non empty set
K460() is non empty L9()
K461() is non empty V107() M20(K460())
K462() is non empty V107() V129() V186() M23(K460(),K461())
K464() is non empty V107() V129() V131() V133() L9()
K465() is non empty V107() V129() V186() M20(K464())
K7(1,1) is V23( RAT ) V23( INT ) V148() V149() V150() V151() set
K6(K7(1,1)) is set
K7(K7(1,1),1) is V23( RAT ) V23( INT ) V148() V149() V150() V151() set
K6(K7(K7(1,1),1)) is set
K7(K7(1,1),REAL) is V148() V149() V150() set
K6(K7(K7(1,1),REAL)) is set
2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
K7(2,2) is V23( RAT ) V23( INT ) V148() V149() V150() V151() set
K7(K7(2,2),REAL) is V148() V149() V150() set
K6(K7(K7(2,2),REAL)) is set
TOP-REAL 2 is non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous RLTopStruct
the carrier of (TOP-REAL 2) is non empty V2() functional set
K7(NAT,REAL) is V148() V149() V150() set
K6(K7(NAT,REAL)) is set
K285( the carrier of (TOP-REAL 2)) is M9( the carrier of (TOP-REAL 2))
K7( the carrier of (TOP-REAL 2),REAL) is V148() V149() V150() set
K6(K7( the carrier of (TOP-REAL 2),REAL)) is set
K6( the carrier of (TOP-REAL 2)) is set
K700() is TopStruct
the carrier of K700() is set
RealSpace is strict MetrStruct
K705() is TopSpace-like T_2 TopStruct
the carrier of K705() is set
K6(K6( the carrier of (TOP-REAL 2))) is set
K636( the carrier of (TOP-REAL 2)) is Element of K6(K6( the carrier of (TOP-REAL 2)))
K7(NAT,K636( the carrier of (TOP-REAL 2))) is set
K6(K7(NAT,K636( the carrier of (TOP-REAL 2)))) is set
K7(COMPLEX,REAL) is V148() V149() V150() set
K6(K7(COMPLEX,REAL)) is set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V158() V159() V160() V161() V162() V163() V164() set
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() Element of NAT
4 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
3 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
proj2 is V19() V22( the carrier of (TOP-REAL 2)) V23( REAL ) Function-like V46( the carrier of (TOP-REAL 2), REAL ) V148() V149() V150() Element of K6(K7( the carrier of (TOP-REAL 2),REAL))
C is V11() real ext-real set
x is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: x is V158() V159() V160() Element of K6(REAL)
x is set
proj2 . x is V11() real ext-real set
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
Cage (C,0) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
RightComp (Cage (C,0)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
Cl (RightComp (Cage (C,0))) is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Upper_Appr C) . x is functional Element of K636( the carrier of (TOP-REAL 2))
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
Cage (C,0) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
RightComp (Cage (C,0)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
Cl (RightComp (Cage (C,0))) is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Lower_Appr C) . x is functional Element of K636( the carrier of (TOP-REAL 2))
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,x))) is non empty functional closed bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc C is non empty functional closed bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
W-min C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-max C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Lower_Arc C is non empty functional closed bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
E-max C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
W-min C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Upper_Appr C) . x is functional Element of K636( the carrier of (TOP-REAL 2))
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Lower_Appr C) . x is functional Element of K636( the carrier of (TOP-REAL 2))
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
North_Arc C is functional Element of K6( the carrier of (TOP-REAL 2))
Upper_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
Cage (C,0) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
RightComp (Cage (C,0)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
Cl (RightComp (Cage (C,0))) is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Upper_Appr C) . x is functional Element of K636( the carrier of (TOP-REAL 2))
Lim_inf (Upper_Appr C) is functional Element of K6( the carrier of (TOP-REAL 2))
South_Arc C is functional Element of K6( the carrier of (TOP-REAL 2))
Lower_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
Cage (C,0) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
RightComp (Cage (C,0)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
Cl (RightComp (Cage (C,0))) is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Lower_Appr C) . x is functional Element of K636( the carrier of (TOP-REAL 2))
Lim_inf (Lower_Appr C) is functional Element of K6( the carrier of (TOP-REAL 2))
dom proj2 is functional Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[1,1] is Element of K7(NAT,NAT)
{1,1} is non empty V158() V159() V160() V161() V162() V163() set
{1} is non empty V158() V159() V160() V161() V162() V163() set
{{1,1},{1}} is non empty set
C is non empty functional Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Indices (Gauge (C,x)) is set
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[1,2] is Element of K7(NAT,NAT)
{1,2} is non empty V158() V159() V160() V161() V162() V163() set
{{1,2},{1}} is non empty set
C is non empty functional Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Indices (Gauge (C,x)) is set
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[2,1] is Element of K7(NAT,NAT)
{2,1} is non empty V158() V159() V160() V161() V162() V163() set
{2} is non empty V158() V159() V160() V161() V162() V163() set
{{2,1},{2}} is non empty set
C is non empty functional Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Indices (Gauge (C,x)) is set
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
C is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[C,e] is Element of K7(NAT,NAT)
{C,e} is non empty V158() V159() V160() V161() V162() V163() set
{C} is non empty V158() V159() V160() V161() V162() V163() set
{{C,e},{C}} is non empty set
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[C,(e + 1)] is Element of K7(NAT,NAT)
{C,(e + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{C,(e + 1)},{C}} is non empty set
A is non empty functional closed bounded compact non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Gauge (A,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Indices (Gauge (A,x)) is set
(Gauge (A,x)) * (C,e) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (A,x)) * (C,(e + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (A,x)) * (C,e)),((Gauge (A,x)) * (C,(e + 1)))) is V11() real ext-real Element of REAL
Gauge (A,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (A,x)) * (C,e) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (A,x)) * (C,(e + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (A,x)) * (C,e)),((Gauge (A,x)) * (C,(e + 1)))) is V11() real ext-real Element of REAL
len (Gauge (A,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
len (Gauge (A,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (A,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (A,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
N-bound A is V11() real ext-real Element of REAL
S-bound A is V11() real ext-real Element of REAL
(N-bound A) - (S-bound A) is V11() real ext-real Element of REAL
Indices (Gauge (A,x)) is set
2 |^ x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((N-bound A) - (S-bound A)) / (2 |^ x) is V11() real ext-real Element of COMPLEX
2 |^ x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((N-bound A) - (S-bound A)) / (2 |^ x) is V11() real ext-real Element of COMPLEX
C is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x is non empty functional closed bounded compact non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Gauge (x,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (x,x)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (x,x)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (x,x)) * (1,1)),((Gauge (x,x)) * (1,2))) is V11() real ext-real Element of REAL
Gauge (x,C) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (x,C)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (x,C)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (x,C)) * (1,1)),((Gauge (x,C)) * (1,2))) is V11() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[1,(1 + 1)] is Element of K7(NAT,NAT)
{1,(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{1,(1 + 1)},{1}} is non empty set
Indices (Gauge (x,x)) is set
C is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
C + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[C,e] is Element of K7(NAT,NAT)
{C,e} is non empty V158() V159() V160() V161() V162() V163() set
{C} is non empty V158() V159() V160() V161() V162() V163() set
{{C,e},{C}} is non empty set
[(C + 1),e] is Element of K7(NAT,NAT)
{(C + 1),e} is non empty V158() V159() V160() V161() V162() V163() set
{(C + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(C + 1),e},{(C + 1)}} is non empty set
A is non empty functional closed bounded compact non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Gauge (A,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Indices (Gauge (A,x)) is set
(Gauge (A,x)) * (C,e) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (A,x)) * ((C + 1),e) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (A,x)) * (C,e)),((Gauge (A,x)) * ((C + 1),e))) is V11() real ext-real Element of REAL
Gauge (A,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (A,x)) * (C,e) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (A,x)) * ((C + 1),e) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (A,x)) * (C,e)),((Gauge (A,x)) * ((C + 1),e))) is V11() real ext-real Element of REAL
len (Gauge (A,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
len (Gauge (A,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
E-bound A is V11() real ext-real Element of REAL
W-bound A is V11() real ext-real Element of REAL
(E-bound A) - (W-bound A) is V11() real ext-real Element of REAL
width (Gauge (A,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (A,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Indices (Gauge (A,x)) is set
2 |^ x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((E-bound A) - (W-bound A)) / (2 |^ x) is V11() real ext-real Element of COMPLEX
2 |^ x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((E-bound A) - (W-bound A)) / (2 |^ x) is V11() real ext-real Element of COMPLEX
C is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x is non empty functional closed bounded compact non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Gauge (x,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (x,x)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (x,x)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (x,x)) * (1,1)),((Gauge (x,x)) * (2,1))) is V11() real ext-real Element of REAL
Gauge (x,C) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (x,C)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (x,C)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (x,C)) * (1,1)),((Gauge (x,C)) * (2,1))) is V11() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[(1 + 1),1] is Element of K7(NAT,NAT)
{(1 + 1),1} is non empty V158() V159() V160() V161() V162() V163() set
{(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(1 + 1),1},{(1 + 1)}} is non empty set
Indices (Gauge (x,x)) is set
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x is V11() real ext-real set
e is V11() real ext-real set
A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,A) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,A)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,A)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (2,1))) is V11() real ext-real Element of REAL
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,(x + 1)) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,(x + 1))) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,(x + 1))) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,(x + 1))) * (1,1)),((Gauge (C,(x + 1))) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,(x + 1))) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,(x + 1))) * (1,1)),((Gauge (C,(x + 1))) * (2,1))) is V11() real ext-real Element of REAL
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,x)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,x)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,x)) * (1,1)),((Gauge (C,x)) * (2,1))) is V11() real ext-real Element of REAL
(Gauge (C,x)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,x)) * (1,1)),((Gauge (C,x)) * (1,2))) is V11() real ext-real Element of REAL
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Center (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
(L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))))) is V11() real ext-real Element of REAL
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of COMPLEX
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
width (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((Gauge (C,x)) * ((Center (Gauge (C,x))),1)) `1 is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
E-bound C is V11() real ext-real Element of REAL
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
((W-bound C) + (E-bound C)) / 2 is V11() real ext-real Element of COMPLEX
(W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of COMPLEX
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))) `1 is V11() real ext-real Element of REAL
(L~ (Cage (C,x))) /\ (L~ (Cage (C,x))) is functional Element of K6( the carrier of (TOP-REAL 2))
((L~ (Cage (C,x))) /\ (L~ (Cage (C,x)))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
r is V11() real ext-real set
r is V11() real ext-real set
(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))))) - r is V11() real ext-real Element of REAL
k is V11() real ext-real set
k is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
proj2 . k is V11() real ext-real Element of REAL
k `1 is V11() real ext-real Element of REAL
GoB (Cage (C,x)) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
((Gauge (C,x)) * ((Center (Gauge (C,x))),1)) `2 is V11() real ext-real Element of REAL
k `2 is V11() real ext-real Element of REAL
((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))) `2 is V11() real ext-real Element of REAL
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Center (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
(L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))))) is V11() real ext-real Element of REAL
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of COMPLEX
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
width (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((Gauge (C,x)) * ((Center (Gauge (C,x))),1)) `1 is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
E-bound C is V11() real ext-real Element of REAL
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
((W-bound C) + (E-bound C)) / 2 is V11() real ext-real Element of COMPLEX
(W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of COMPLEX
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))) `1 is V11() real ext-real Element of REAL
(L~ (Cage (C,x))) /\ (L~ (Cage (C,x))) is functional Element of K6( the carrier of (TOP-REAL 2))
((L~ (Cage (C,x))) /\ (L~ (Cage (C,x)))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
r is V11() real ext-real set
r is V11() real ext-real set
(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))))) + r is V11() real ext-real Element of REAL
k is V11() real ext-real set
k is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
proj2 . k is V11() real ext-real Element of REAL
k `1 is V11() real ext-real Element of REAL
GoB (Cage (C,x)) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
((Gauge (C,x)) * ((Center (Gauge (C,x))),1)) `2 is V11() real ext-real Element of REAL
k `2 is V11() real ext-real Element of REAL
((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))) `2 is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of COMPLEX
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Center (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
(L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
LMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of COMPLEX
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Center (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
(L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UMP C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound C) + (W-bound C)) / 2),(upper_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP C) `2 is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
((W-bound C) + (E-bound C)) / 2 is V11() real ext-real Element of COMPLEX
(UMP C) `1 is V11() real ext-real Element of REAL
|[((UMP C) `1),((UMP C) `2)]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (L~ (Cage (C,x)))) `1 is V11() real ext-real Element of REAL
|[((UMP (L~ (Cage (C,x)))) `1),((UMP (L~ (Cage (C,x)))) `2)]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of COMPLEX
Vertical_Line (((W-bound C) + (E-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((W-bound C) + (E-bound C)) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)))) is V11() real ext-real Element of REAL
north_halfline (UMP (L~ (Cage (C,x)))) is non empty functional connected convex Element of K6( the carrier of (TOP-REAL 2))
{(UMP (L~ (Cage (C,x))))} is non empty functional Element of K6( the carrier of (TOP-REAL 2))
(north_halfline (UMP (L~ (Cage (C,x))))) \ {(UMP (L~ (Cage (C,x))))} is functional Element of K6( the carrier of (TOP-REAL 2))
m is set
OO is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
OO `2 is V11() real ext-real Element of REAL
OO `1 is V11() real ext-real Element of REAL
proj2 . OO is V11() real ext-real Element of REAL
UBD (L~ (Cage (C,x))) is non empty functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
BDD (L~ (Cage (C,x))) is functional open Element of K6( the carrier of (TOP-REAL 2))
UBD C is non empty functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
LMP C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound C) + (W-bound C)) / 2),(lower_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP C) `2 is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
LMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
((W-bound C) + (E-bound C)) / 2 is V11() real ext-real Element of COMPLEX
(LMP C) `1 is V11() real ext-real Element of REAL
|[((LMP C) `1),((LMP C) `2)]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (L~ (Cage (C,x)))) `1 is V11() real ext-real Element of REAL
|[((LMP (L~ (Cage (C,x)))) `1),((LMP (L~ (Cage (C,x)))) `2)]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of COMPLEX
Vertical_Line (((W-bound C) + (E-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((W-bound C) + (E-bound C)) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)))) is V11() real ext-real Element of REAL
south_halfline (LMP (L~ (Cage (C,x)))) is non empty functional connected convex Element of K6( the carrier of (TOP-REAL 2))
{(LMP (L~ (Cage (C,x))))} is non empty functional Element of K6( the carrier of (TOP-REAL 2))
(south_halfline (LMP (L~ (Cage (C,x))))) \ {(LMP (L~ (Cage (C,x))))} is functional Element of K6( the carrier of (TOP-REAL 2))
m is set
OO is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
OO `2 is V11() real ext-real Element of REAL
OO `1 is V11() real ext-real Element of REAL
proj2 . OO is V11() real ext-real Element of REAL
UBD (L~ (Cage (C,x))) is non empty functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
BDD (L~ (Cage (C,x))) is functional open Element of K6( the carrier of (TOP-REAL 2))
UBD C is non empty functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Center (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of COMPLEX
(L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
[(Center (Gauge (C,x))),(len (Gauge (C,x)))] is Element of K7(NAT,NAT)
{(Center (Gauge (C,x))),(len (Gauge (C,x)))} is non empty V158() V159() V160() V161() V162() V163() set
{(Center (Gauge (C,x)))} is non empty V158() V159() V160() V161() V162() V163() set
{{(Center (Gauge (C,x))),(len (Gauge (C,x)))},{(Center (Gauge (C,x)))}} is non empty set
Indices (Gauge (C,x)) is set
[(Center (Gauge (C,x))),1] is Element of K7(NAT,NAT)
{(Center (Gauge (C,x))),1} is non empty V158() V159() V160() V161() V162() V163() set
{{(Center (Gauge (C,x))),1},{(Center (Gauge (C,x)))}} is non empty set
(LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) /\ (L~ (Cage (C,x))) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) /\ (L~ (Cage (C,x)))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) /\ (L~ (Cage (C,x))))) is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),m) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
((Gauge (C,x)) * ((Center (Gauge (C,x))),m)) `2 is V11() real ext-real Element of REAL
(UMP (L~ (Cage (C,x)))) `1 is V11() real ext-real Element of REAL
((Gauge (C,x)) * ((Center (Gauge (C,x))),m)) `1 is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
E-bound C is V11() real ext-real Element of REAL
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
((W-bound C) + (E-bound C)) / 2 is V11() real ext-real Element of COMPLEX
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
LMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Center (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of COMPLEX
(L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))))))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
[(Center (Gauge (C,x))),(len (Gauge (C,x)))] is Element of K7(NAT,NAT)
{(Center (Gauge (C,x))),(len (Gauge (C,x)))} is non empty V158() V159() V160() V161() V162() V163() set
{(Center (Gauge (C,x)))} is non empty V158() V159() V160() V161() V162() V163() set
{{(Center (Gauge (C,x))),(len (Gauge (C,x)))},{(Center (Gauge (C,x)))}} is non empty set
Indices (Gauge (C,x)) is set
[(Center (Gauge (C,x))),1] is Element of K7(NAT,NAT)
{(Center (Gauge (C,x))),1} is non empty V158() V159() V160() V161() V162() V163() set
{{(Center (Gauge (C,x))),1},{(Center (Gauge (C,x)))}} is non empty set
(LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) /\ (L~ (Cage (C,x))) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) /\ (L~ (Cage (C,x)))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((LSeg (((Gauge (C,x)) * ((Center (Gauge (C,x))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))))) /\ (L~ (Cage (C,x))))) is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),m) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
((Gauge (C,x)) * ((Center (Gauge (C,x))),m)) `2 is V11() real ext-real Element of REAL
(LMP (L~ (Cage (C,x)))) `1 is V11() real ext-real Element of REAL
((Gauge (C,x)) * ((Center (Gauge (C,x))),m)) `1 is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
E-bound C is V11() real ext-real Element of REAL
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
((W-bound C) + (E-bound C)) / 2 is V11() real ext-real Element of COMPLEX
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of COMPLEX
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
(W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(x -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,1) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Center (Gauge (C,1)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
len (Gauge (C,1)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Center (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Lower_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
(Upper_Arc (L~ (Cage (C,x)))) \/ (Lower_Arc (L~ (Cage (C,x)))) is non empty functional Element of K6( the carrier of (TOP-REAL 2))
width (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),A) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,1)) * ((Center (Gauge (C,1))),(len (Gauge (C,1)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),(len (Gauge (C,1))))),((Gauge (C,x)) * ((Center (Gauge (C,x))),A))) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
width (Gauge (C,1)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
k is set
((Gauge (C,1)) * ((Center (Gauge (C,1))),(len (Gauge (C,1))))) `1 is V11() real ext-real Element of REAL
((Gauge (C,x)) * ((Center (Gauge (C,x))),A)) `1 is V11() real ext-real Element of REAL
k is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
k `1 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
(Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
k `2 is V11() real ext-real Element of REAL
(UMP (Upper_Arc (L~ (Cage (C,x))))) `2 is V11() real ext-real Element of REAL
(Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
((Gauge (C,x)) * ((Center (Gauge (C,x))),(len (Gauge (C,x))))) `2 is V11() real ext-real Element of REAL
((Gauge (C,1)) * ((Center (Gauge (C,1))),(len (Gauge (C,1))))) `2 is V11() real ext-real Element of REAL
((Gauge (C,x)) * ((Center (Gauge (C,x))),A)) `2 is V11() real ext-real Element of REAL
(UMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
LMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Lower_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of COMPLEX
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
(W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
x -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(x -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,1) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
Center (Gauge (C,1)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
len (Gauge (C,1)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,x) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
len (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
(Upper_Arc (L~ (Cage (C,x)))) \/ (Lower_Arc (L~ (Cage (C,x)))) is non empty functional Element of K6( the carrier of (TOP-REAL 2))
Center (Gauge (C,x)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,x)) * ((Center (Gauge (C,x))),A) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,1)) * ((Center (Gauge (C,1))),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,x)) * ((Center (Gauge (C,x))),A))) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
k is set
((Gauge (C,1)) * ((Center (Gauge (C,1))),1)) `1 is V11() real ext-real Element of REAL
((Gauge (C,x)) * ((Center (Gauge (C,x))),A)) `1 is V11() real ext-real Element of REAL
k is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
k `1 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
(Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
(LMP (Lower_Arc (L~ (Cage (C,x))))) `2 is V11() real ext-real Element of REAL
k `2 is V11() real ext-real Element of REAL
((Gauge (C,1)) * ((Center (Gauge (C,1))),1)) `2 is V11() real ext-real Element of REAL
(Gauge (C,x)) * ((Center (Gauge (C,x))),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
((Gauge (C,x)) * ((Center (Gauge (C,x))),1)) `2 is V11() real ext-real Element of REAL
((Gauge (C,x)) * ((Center (Gauge (C,x))),A)) `2 is V11() real ext-real Element of REAL
(LMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UMP C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound C) + (W-bound C)) / 2),(upper_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP C) `2 is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,x))))) `2 is V11() real ext-real Element of REAL
UMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
LMP C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound C) + (W-bound C)) / 2),(lower_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP C) `2 is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,x))))) `2 is V11() real ext-real Element of REAL
LMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of COMPLEX
LeftComp (Cage (C,x)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
LeftComp (Cage (C,x)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
(W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
((UMP (L~ (Cage (C,x)))) `2) - ((UMP (L~ (Cage (C,x)))) `2) is V11() real ext-real Element of REAL
(W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))))) - (((UMP (L~ (Cage (C,x)))) `2) - ((UMP (L~ (Cage (C,x)))) `2)) is V11() real ext-real Element of REAL
OO is V11() real ext-real set
r is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
proj2 . r is V11() real ext-real Element of REAL
r `2 is V11() real ext-real Element of REAL
north_halfline r is non empty functional connected convex Element of K6( the carrier of (TOP-REAL 2))
k is set
k is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
k `1 is V11() real ext-real Element of REAL
r `1 is V11() real ext-real Element of REAL
proj2 . k is V11() real ext-real Element of REAL
k `2 is V11() real ext-real Element of REAL
UBD (L~ (Cage (C,x))) is non empty functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
LMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
LMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of COMPLEX
LeftComp (Cage (C,x)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
LeftComp (Cage (C,x)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
(W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
((LMP (L~ (Cage (C,x)))) `2) - ((LMP (L~ (Cage (C,x)))) `2) is V11() real ext-real Element of REAL
(W-bound (L~ (Cage (C,x)))) + (E-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))))) + (((LMP (L~ (Cage (C,x)))) `2) - ((LMP (L~ (Cage (C,x)))) `2)) is V11() real ext-real Element of REAL
OO is V11() real ext-real set
r is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
proj2 . r is V11() real ext-real Element of REAL
r `2 is V11() real ext-real Element of REAL
south_halfline r is non empty functional connected convex Element of K6( the carrier of (TOP-REAL 2))
k is set
k is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
k `1 is V11() real ext-real Element of REAL
r `1 is V11() real ext-real Element of REAL
proj2 . k is V11() real ext-real Element of REAL
k `2 is V11() real ext-real Element of REAL
UBD (L~ (Cage (C,x))) is non empty functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,x))))) `2 is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,x))))) + (W-bound (Upper_Arc (L~ (Cage (C,x)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,x))))) `2 is V11() real ext-real Element of REAL
UMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
UMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,x))))) `2 is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,x) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,x)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,x))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,x)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,x)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,x))))) + (W-bound (Lower_Arc (L~ (Cage (C,x)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,x))))) `2 is V11() real ext-real Element of REAL
LMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
LMP (L~ (Cage (C,x))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,x))) is V11() real ext-real Element of REAL
(E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x)))) is V11() real ext-real Element of REAL
((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2 } is set
(L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,x))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,x)))) + (W-bound (L~ (Cage (C,x))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (L~ (Cage (C,x)))) `2 is V11() real ext-real Element of REAL
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
W-min C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
North_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
Euclid 2 is non empty strict Reflexive discerning symmetric triangle MetrStruct
the carrier of (Euclid 2) is non empty set
Upper_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
x is Element of the carrier of (Euclid 2)
x is V11() real ext-real set
Ball (x,x) is Element of K6( the carrier of (Euclid 2))
K6( the carrier of (Euclid 2)) is set
x / 2 is V11() real ext-real Element of COMPLEX
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,e) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,e)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,e)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,e)) * (1,1)),((Gauge (C,e)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,e)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,e)) * (1,1)),((Gauge (C,e)) * (2,1))) is V11() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Upper_Appr C) . A is functional Element of K636( the carrier of (TOP-REAL 2))
Gauge (C,A) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,A)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,A)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (2,1))) is V11() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,A) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,A)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
W-min (L~ (Cage (C,A))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A))))) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
len (Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
left_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
(left_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1)) /\ (right_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1)) is functional Element of K6( the carrier of (TOP-REAL 2))
LSeg ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,A))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
rng (Cage (C,A)) is V2() functional Element of K6( the carrier of (TOP-REAL 2))
(Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))) /. 1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
L~ (Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
W-min (L~ (Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A))))))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Indices (Gauge (C,A)) is set
(Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))) /. (1 + 1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
OO is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[OO,r] is Element of K7(NAT,NAT)
{OO,r} is non empty V158() V159() V160() V161() V162() V163() set
{OO} is non empty V158() V159() V160() V161() V162() V163() set
{{OO,r},{OO}} is non empty set
r + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[OO,(r + 1)] is Element of K7(NAT,NAT)
{OO,(r + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{OO,(r + 1)},{OO}} is non empty set
(Gauge (C,A)) * (OO,r) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * (OO,(r + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
len (Gauge (C,A)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
OO + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (C,A)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[(OO + 1),r] is Element of K7(NAT,NAT)
{(OO + 1),r} is non empty V158() V159() V160() V161() V162() V163() set
{(OO + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(OO + 1),r},{(OO + 1)}} is non empty set
[(1 + 1),1] is Element of K7(NAT,NAT)
{(1 + 1),1} is non empty V158() V159() V160() V161() V162() V163() set
{(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(1 + 1),1},{(1 + 1)}} is non empty set
(Gauge (C,A)) * ((1 + 1),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * ((1 + 1),1))) is V11() real ext-real Element of REAL
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) - (W-bound C) is V11() real ext-real Element of REAL
2 |^ A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((E-bound C) - (W-bound C)) / (2 |^ A) is V11() real ext-real Element of COMPLEX
(Gauge (C,A)) * ((OO + 1),r) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (OO,r)),((Gauge (C,A)) * ((OO + 1),r))) is V11() real ext-real Element of REAL
((Gauge (C,A)) * ((OO + 1),r)) `1 is V11() real ext-real Element of REAL
((Gauge (C,A)) * (OO,r)) `1 is V11() real ext-real Element of REAL
(((Gauge (C,A)) * ((OO + 1),r)) `1) - (((Gauge (C,A)) * (OO,r)) `1) is V11() real ext-real Element of REAL
[1,(1 + 1)] is Element of K7(NAT,NAT)
{1,(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{1,(1 + 1)},{1}} is non empty set
(Gauge (C,A)) * (1,(1 + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (1,(1 + 1)))) is V11() real ext-real Element of REAL
N-bound C is V11() real ext-real Element of REAL
S-bound C is V11() real ext-real Element of REAL
(N-bound C) - (S-bound C) is V11() real ext-real Element of REAL
((N-bound C) - (S-bound C)) / (2 |^ A) is V11() real ext-real Element of COMPLEX
dist (((Gauge (C,A)) * (OO,r)),((Gauge (C,A)) * (OO,(r + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,A)) * (OO,(r + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,A)) * (OO,r)) `2 is V11() real ext-real Element of REAL
(((Gauge (C,A)) * (OO,(r + 1))) `2) - (((Gauge (C,A)) * (OO,r)) `2) is V11() real ext-real Element of REAL
(x / 2) + (x / 2) is V11() real ext-real Element of COMPLEX
((((Gauge (C,A)) * ((OO + 1),r)) `1) - (((Gauge (C,A)) * (OO,r)) `1)) + ((((Gauge (C,A)) * (OO,(r + 1))) `2) - (((Gauge (C,A)) * (OO,r)) `2)) is V11() real ext-real Element of REAL
GoB (Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1,(GoB (Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
GoB (Cage (C,A)) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1,(GoB (Cage (C,A)))) is functional Element of K6( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1,(Gauge (C,A))) is functional Element of K6( the carrier of (TOP-REAL 2))
cell ((Gauge (C,A)),OO,r) is functional Element of K6( the carrier of (TOP-REAL 2))
(W-min (L~ (Cage (C,A)))) `1 is V11() real ext-real Element of REAL
(W-min C) `1 is V11() real ext-real Element of REAL
(W-min (L~ (Cage (C,A)))) `2 is V11() real ext-real Element of REAL
(W-min C) `2 is V11() real ext-real Element of REAL
dist ((W-min C),(W-min (L~ (Cage (C,A))))) is V11() real ext-real Element of REAL
m is Element of the carrier of (Euclid 2)
dist (x,m) is V11() real ext-real Element of REAL
Lim_inf (Upper_Appr C) is functional Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
E-max C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
North_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
Euclid 2 is non empty strict Reflexive discerning symmetric triangle MetrStruct
the carrier of (Euclid 2) is non empty set
Upper_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
x is Element of the carrier of (Euclid 2)
x is V11() real ext-real set
Ball (x,x) is Element of K6( the carrier of (Euclid 2))
K6( the carrier of (Euclid 2)) is set
x / 2 is V11() real ext-real Element of COMPLEX
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,e) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,e)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,e)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,e)) * (1,1)),((Gauge (C,e)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,e)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,e)) * (1,1)),((Gauge (C,e)) * (2,1))) is V11() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Upper_Appr C) . A is functional Element of K636( the carrier of (TOP-REAL 2))
Gauge (C,A) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,A)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,A)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (2,1))) is V11() real ext-real Element of REAL
Cage (C,A) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,A)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
E-max (L~ (Cage (C,A))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Upper_Arc (L~ (Cage (C,A))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A))))) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
len (Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
left_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
(left_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1)) /\ (right_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1)) is functional Element of K6( the carrier of (TOP-REAL 2))
LSeg ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
rng (Cage (C,A)) is V2() functional Element of K6( the carrier of (TOP-REAL 2))
(Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))) /. 1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
L~ (Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
E-max (L~ (Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A))))))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Indices (Gauge (C,A)) is set
(Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))) /. (1 + 1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
OO is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
r + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[OO,(r + 1)] is Element of K7(NAT,NAT)
{OO,(r + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{OO} is non empty V158() V159() V160() V161() V162() V163() set
{{OO,(r + 1)},{OO}} is non empty set
[OO,r] is Element of K7(NAT,NAT)
{OO,r} is non empty V158() V159() V160() V161() V162() V163() set
{{OO,r},{OO}} is non empty set
(Gauge (C,A)) * (OO,(r + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * (OO,r) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
len (Gauge (C,A)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
OO - 1 is V11() real ext-real V32() Element of REAL
1 - 1 is V11() real ext-real V32() Element of REAL
OO -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(OO -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
GoB (Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1,(GoB (Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
GoB (Cage (C,A)) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1,(GoB (Cage (C,A)))) is functional Element of K6( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1,(Gauge (C,A))) is functional Element of K6( the carrier of (TOP-REAL 2))
cell ((Gauge (C,A)),(OO -' 1),r) is functional Element of K6( the carrier of (TOP-REAL 2))
width (Gauge (C,A)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[(OO -' 1),(r + 1)] is Element of K7(NAT,NAT)
{(OO -' 1),(r + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{(OO -' 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(OO -' 1),(r + 1)},{(OO -' 1)}} is non empty set
[(OO -' 1),r] is Element of K7(NAT,NAT)
{(OO -' 1),r} is non empty V158() V159() V160() V161() V162() V163() set
{{(OO -' 1),r},{(OO -' 1)}} is non empty set
[1,(1 + 1)] is Element of K7(NAT,NAT)
{1,(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{1,(1 + 1)},{1}} is non empty set
(Gauge (C,A)) * (1,(1 + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (1,(1 + 1)))) is V11() real ext-real Element of REAL
N-bound C is V11() real ext-real Element of REAL
S-bound C is V11() real ext-real Element of REAL
(N-bound C) - (S-bound C) is V11() real ext-real Element of REAL
2 |^ A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((N-bound C) - (S-bound C)) / (2 |^ A) is V11() real ext-real Element of COMPLEX
(Gauge (C,A)) * ((OO -' 1),r) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * ((OO -' 1),(r + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * ((OO -' 1),r)),((Gauge (C,A)) * ((OO -' 1),(r + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,A)) * ((OO -' 1),(r + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,A)) * ((OO -' 1),r)) `2 is V11() real ext-real Element of REAL
(((Gauge (C,A)) * ((OO -' 1),(r + 1))) `2) - (((Gauge (C,A)) * ((OO -' 1),r)) `2) is V11() real ext-real Element of REAL
[(1 + 1),1] is Element of K7(NAT,NAT)
{(1 + 1),1} is non empty V158() V159() V160() V161() V162() V163() set
{(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(1 + 1),1},{(1 + 1)}} is non empty set
(Gauge (C,A)) * ((1 + 1),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * ((1 + 1),1))) is V11() real ext-real Element of REAL
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) - (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) - (W-bound C)) / (2 |^ A) is V11() real ext-real Element of COMPLEX
(Gauge (C,A)) * (((OO -' 1) + 1),r) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * ((OO -' 1),r)),((Gauge (C,A)) * (((OO -' 1) + 1),r))) is V11() real ext-real Element of REAL
((Gauge (C,A)) * (((OO -' 1) + 1),r)) `1 is V11() real ext-real Element of REAL
((Gauge (C,A)) * ((OO -' 1),r)) `1 is V11() real ext-real Element of REAL
(((Gauge (C,A)) * (((OO -' 1) + 1),r)) `1) - (((Gauge (C,A)) * ((OO -' 1),r)) `1) is V11() real ext-real Element of REAL
(x / 2) + (x / 2) is V11() real ext-real Element of COMPLEX
((((Gauge (C,A)) * (((OO -' 1) + 1),r)) `1) - (((Gauge (C,A)) * ((OO -' 1),r)) `1)) + ((((Gauge (C,A)) * ((OO -' 1),(r + 1))) `2) - (((Gauge (C,A)) * ((OO -' 1),r)) `2)) is V11() real ext-real Element of REAL
(E-max C) `1 is V11() real ext-real Element of REAL
(E-max (L~ (Cage (C,A)))) `1 is V11() real ext-real Element of REAL
(E-max (L~ (Cage (C,A)))) `2 is V11() real ext-real Element of REAL
(E-max C) `2 is V11() real ext-real Element of REAL
dist ((E-max C),(E-max (L~ (Cage (C,A))))) is V11() real ext-real Element of REAL
m is Element of the carrier of (Euclid 2)
dist (x,m) is V11() real ext-real Element of REAL
Lim_inf (Upper_Appr C) is functional Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
W-min C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
South_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
Euclid 2 is non empty strict Reflexive discerning symmetric triangle MetrStruct
the carrier of (Euclid 2) is non empty set
Lower_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
x is Element of the carrier of (Euclid 2)
x is V11() real ext-real set
Ball (x,x) is Element of K6( the carrier of (Euclid 2))
K6( the carrier of (Euclid 2)) is set
x / 2 is V11() real ext-real Element of COMPLEX
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,e) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,e)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,e)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,e)) * (1,1)),((Gauge (C,e)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,e)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,e)) * (1,1)),((Gauge (C,e)) * (2,1))) is V11() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Lower_Appr C) . A is functional Element of K636( the carrier of (TOP-REAL 2))
Gauge (C,A) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,A)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,A)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (2,1))) is V11() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,A) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,A)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
W-min (L~ (Cage (C,A))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A))))) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
len (Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
left_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
(left_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1)) /\ (right_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1)) is functional Element of K6( the carrier of (TOP-REAL 2))
LSeg ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,A))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
rng (Cage (C,A)) is V2() functional Element of K6( the carrier of (TOP-REAL 2))
(Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))) /. 1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
L~ (Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
W-min (L~ (Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A))))))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Indices (Gauge (C,A)) is set
(Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))) /. (1 + 1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
OO is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[OO,r] is Element of K7(NAT,NAT)
{OO,r} is non empty V158() V159() V160() V161() V162() V163() set
{OO} is non empty V158() V159() V160() V161() V162() V163() set
{{OO,r},{OO}} is non empty set
r + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[OO,(r + 1)] is Element of K7(NAT,NAT)
{OO,(r + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{OO,(r + 1)},{OO}} is non empty set
(Gauge (C,A)) * (OO,r) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * (OO,(r + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
len (Gauge (C,A)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
OO + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
width (Gauge (C,A)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[(OO + 1),r] is Element of K7(NAT,NAT)
{(OO + 1),r} is non empty V158() V159() V160() V161() V162() V163() set
{(OO + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(OO + 1),r},{(OO + 1)}} is non empty set
[(1 + 1),1] is Element of K7(NAT,NAT)
{(1 + 1),1} is non empty V158() V159() V160() V161() V162() V163() set
{(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(1 + 1),1},{(1 + 1)}} is non empty set
(Gauge (C,A)) * ((1 + 1),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * ((1 + 1),1))) is V11() real ext-real Element of REAL
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) - (W-bound C) is V11() real ext-real Element of REAL
2 |^ A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((E-bound C) - (W-bound C)) / (2 |^ A) is V11() real ext-real Element of COMPLEX
(Gauge (C,A)) * ((OO + 1),r) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (OO,r)),((Gauge (C,A)) * ((OO + 1),r))) is V11() real ext-real Element of REAL
((Gauge (C,A)) * ((OO + 1),r)) `1 is V11() real ext-real Element of REAL
((Gauge (C,A)) * (OO,r)) `1 is V11() real ext-real Element of REAL
(((Gauge (C,A)) * ((OO + 1),r)) `1) - (((Gauge (C,A)) * (OO,r)) `1) is V11() real ext-real Element of REAL
[1,(1 + 1)] is Element of K7(NAT,NAT)
{1,(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{1,(1 + 1)},{1}} is non empty set
(Gauge (C,A)) * (1,(1 + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (1,(1 + 1)))) is V11() real ext-real Element of REAL
N-bound C is V11() real ext-real Element of REAL
S-bound C is V11() real ext-real Element of REAL
(N-bound C) - (S-bound C) is V11() real ext-real Element of REAL
((N-bound C) - (S-bound C)) / (2 |^ A) is V11() real ext-real Element of COMPLEX
dist (((Gauge (C,A)) * (OO,r)),((Gauge (C,A)) * (OO,(r + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,A)) * (OO,(r + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,A)) * (OO,r)) `2 is V11() real ext-real Element of REAL
(((Gauge (C,A)) * (OO,(r + 1))) `2) - (((Gauge (C,A)) * (OO,r)) `2) is V11() real ext-real Element of REAL
(x / 2) + (x / 2) is V11() real ext-real Element of COMPLEX
((((Gauge (C,A)) * ((OO + 1),r)) `1) - (((Gauge (C,A)) * (OO,r)) `1)) + ((((Gauge (C,A)) * (OO,(r + 1))) `2) - (((Gauge (C,A)) * (OO,r)) `2)) is V11() real ext-real Element of REAL
GoB (Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1,(GoB (Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
GoB (Cage (C,A)) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1,(GoB (Cage (C,A)))) is functional Element of K6( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(W-min (L~ (Cage (C,A)))))),1,(Gauge (C,A))) is functional Element of K6( the carrier of (TOP-REAL 2))
cell ((Gauge (C,A)),OO,r) is functional Element of K6( the carrier of (TOP-REAL 2))
(W-min (L~ (Cage (C,A)))) `1 is V11() real ext-real Element of REAL
(W-min C) `1 is V11() real ext-real Element of REAL
(W-min (L~ (Cage (C,A)))) `2 is V11() real ext-real Element of REAL
(W-min C) `2 is V11() real ext-real Element of REAL
dist ((W-min C),(W-min (L~ (Cage (C,A))))) is V11() real ext-real Element of REAL
m is Element of the carrier of (Euclid 2)
dist (x,m) is V11() real ext-real Element of REAL
Lim_inf (Lower_Appr C) is functional Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
E-max C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
South_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
Euclid 2 is non empty strict Reflexive discerning symmetric triangle MetrStruct
the carrier of (Euclid 2) is non empty set
Lower_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
x is Element of the carrier of (Euclid 2)
x is V11() real ext-real set
Ball (x,x) is Element of K6( the carrier of (Euclid 2))
K6( the carrier of (Euclid 2)) is set
x / 2 is V11() real ext-real Element of COMPLEX
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,e) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,e)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,e)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,e)) * (1,1)),((Gauge (C,e)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,e)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,e)) * (1,1)),((Gauge (C,e)) * (2,1))) is V11() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Lower_Appr C) . A is functional Element of K636( the carrier of (TOP-REAL 2))
Gauge (C,A) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,A)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,A)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (2,1))) is V11() real ext-real Element of REAL
Cage (C,A) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,A)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
E-max (L~ (Cage (C,A))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Lower_Arc (L~ (Cage (C,A))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A))))) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
len (Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
left_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
(left_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1)) /\ (right_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1)) is functional Element of K6( the carrier of (TOP-REAL 2))
LSeg ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1) is functional Element of K6( the carrier of (TOP-REAL 2))
rng (Cage (C,A)) is V2() functional Element of K6( the carrier of (TOP-REAL 2))
(Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))) /. 1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
L~ (Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
E-max (L~ (Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A))))))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Indices (Gauge (C,A)) is set
(Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))) /. (1 + 1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
OO is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
r + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[OO,(r + 1)] is Element of K7(NAT,NAT)
{OO,(r + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{OO} is non empty V158() V159() V160() V161() V162() V163() set
{{OO,(r + 1)},{OO}} is non empty set
[OO,r] is Element of K7(NAT,NAT)
{OO,r} is non empty V158() V159() V160() V161() V162() V163() set
{{OO,r},{OO}} is non empty set
(Gauge (C,A)) * (OO,(r + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * (OO,r) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
len (Gauge (C,A)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
OO - 1 is V11() real ext-real V32() Element of REAL
1 - 1 is V11() real ext-real V32() Element of REAL
OO -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(OO -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
GoB (Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1,(GoB (Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))))) is functional Element of K6( the carrier of (TOP-REAL 2))
GoB (Cage (C,A)) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1,(GoB (Cage (C,A)))) is functional Element of K6( the carrier of (TOP-REAL 2))
right_cell ((Rotate ((Cage (C,A)),(E-max (L~ (Cage (C,A)))))),1,(Gauge (C,A))) is functional Element of K6( the carrier of (TOP-REAL 2))
cell ((Gauge (C,A)),(OO -' 1),r) is functional Element of K6( the carrier of (TOP-REAL 2))
width (Gauge (C,A)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[(OO -' 1),(r + 1)] is Element of K7(NAT,NAT)
{(OO -' 1),(r + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{(OO -' 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(OO -' 1),(r + 1)},{(OO -' 1)}} is non empty set
[(OO -' 1),r] is Element of K7(NAT,NAT)
{(OO -' 1),r} is non empty V158() V159() V160() V161() V162() V163() set
{{(OO -' 1),r},{(OO -' 1)}} is non empty set
[1,(1 + 1)] is Element of K7(NAT,NAT)
{1,(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{1,(1 + 1)},{1}} is non empty set
(Gauge (C,A)) * (1,(1 + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * (1,(1 + 1)))) is V11() real ext-real Element of REAL
N-bound C is V11() real ext-real Element of REAL
S-bound C is V11() real ext-real Element of REAL
(N-bound C) - (S-bound C) is V11() real ext-real Element of REAL
2 |^ A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((N-bound C) - (S-bound C)) / (2 |^ A) is V11() real ext-real Element of COMPLEX
(Gauge (C,A)) * ((OO -' 1),r) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,A)) * ((OO -' 1),(r + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * ((OO -' 1),r)),((Gauge (C,A)) * ((OO -' 1),(r + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,A)) * ((OO -' 1),(r + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,A)) * ((OO -' 1),r)) `2 is V11() real ext-real Element of REAL
(((Gauge (C,A)) * ((OO -' 1),(r + 1))) `2) - (((Gauge (C,A)) * ((OO -' 1),r)) `2) is V11() real ext-real Element of REAL
[(1 + 1),1] is Element of K7(NAT,NAT)
{(1 + 1),1} is non empty V158() V159() V160() V161() V162() V163() set
{(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(1 + 1),1},{(1 + 1)}} is non empty set
(Gauge (C,A)) * ((1 + 1),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * (1,1)),((Gauge (C,A)) * ((1 + 1),1))) is V11() real ext-real Element of REAL
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) - (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) - (W-bound C)) / (2 |^ A) is V11() real ext-real Element of COMPLEX
(Gauge (C,A)) * (((OO -' 1) + 1),r) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,A)) * ((OO -' 1),r)),((Gauge (C,A)) * (((OO -' 1) + 1),r))) is V11() real ext-real Element of REAL
((Gauge (C,A)) * (((OO -' 1) + 1),r)) `1 is V11() real ext-real Element of REAL
((Gauge (C,A)) * ((OO -' 1),r)) `1 is V11() real ext-real Element of REAL
(((Gauge (C,A)) * (((OO -' 1) + 1),r)) `1) - (((Gauge (C,A)) * ((OO -' 1),r)) `1) is V11() real ext-real Element of REAL
(x / 2) + (x / 2) is V11() real ext-real Element of COMPLEX
((((Gauge (C,A)) * (((OO -' 1) + 1),r)) `1) - (((Gauge (C,A)) * ((OO -' 1),r)) `1)) + ((((Gauge (C,A)) * ((OO -' 1),(r + 1))) `2) - (((Gauge (C,A)) * ((OO -' 1),r)) `2)) is V11() real ext-real Element of REAL
(E-max C) `1 is V11() real ext-real Element of REAL
(E-max (L~ (Cage (C,A)))) `1 is V11() real ext-real Element of REAL
(E-max (L~ (Cage (C,A)))) `2 is V11() real ext-real Element of REAL
(E-max C) `2 is V11() real ext-real Element of REAL
dist ((E-max C),(E-max (L~ (Cage (C,A))))) is V11() real ext-real Element of REAL
m is Element of the carrier of (Euclid 2)
dist (x,m) is V11() real ext-real Element of REAL
Lim_inf (Lower_Appr C) is functional Element of K6( the carrier of (TOP-REAL 2))
the topology of (TOP-REAL 2) is Element of K6(K6( the carrier of (TOP-REAL 2)))
TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) is strict TopStruct
Euclid 2 is non empty strict Reflexive discerning symmetric triangle MetrStruct
TopSpaceMetr (Euclid 2) is TopStruct
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UMP C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound C) + (W-bound C)) / 2),(upper_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
North_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
((W-bound C) + (E-bound C)) / 2 is V11() real ext-real Element of COMPLEX
{ (UMP (Upper_Arc (L~ (Cage (C,b1))))) where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT : not b1 <= 0 } is set
Cage (C,1) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,1)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
N-bound (L~ (Cage (C,1))) is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
N-bound C is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(N-bound C)]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
|[(((W-bound C) + (E-bound C)) / 2),(N-bound C)]| `1 is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]| `1 is V11() real ext-real Element of REAL
OO is set
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,r) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,r)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,r))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,r)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,r)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,r)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,r))))) + (W-bound (Upper_Arc (L~ (Cage (C,r))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,r))))) + (W-bound (Upper_Arc (L~ (Cage (C,r)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,r))))) + (W-bound (Upper_Arc (L~ (Cage (C,r)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,r))))) + (W-bound (Upper_Arc (L~ (Cage (C,r)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,r))))) + (W-bound (Upper_Arc (L~ (Cage (C,r)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,r))))) + (W-bound (Upper_Arc (L~ (Cage (C,r)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,r))))) + (W-bound (Upper_Arc (L~ (Cage (C,r)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,r))))) + (W-bound (Upper_Arc (L~ (Cage (C,r)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,r))))) + (W-bound (Upper_Arc (L~ (Cage (C,r)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
OO is functional Element of K6( the carrier of (TOP-REAL 2))
(LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO)) is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO)))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg (|[(((W-bound C) + (E-bound C)) / 2),(lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO)))]|,|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
|[(((W-bound C) + (E-bound C)) / 2),(N-bound C)]| `2 is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]| `2 is V11() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,k) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,k)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,k))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,k)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,k)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,k)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,k))))) `1 is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,k))) is V11() real ext-real Element of REAL
E-bound (L~ (Cage (C,k))) is V11() real ext-real Element of REAL
(W-bound (L~ (Cage (C,k)))) + (E-bound (L~ (Cage (C,k)))) is V11() real ext-real Element of REAL
(W-bound (Upper_Arc (L~ (Cage (C,k))))) + (E-bound (Upper_Arc (L~ (Cage (C,k))))) is V11() real ext-real Element of REAL
((W-bound (Upper_Arc (L~ (Cage (C,k))))) + (E-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2 is V11() real ext-real Element of COMPLEX
(W-bound (L~ (Cage (C,k)))) + (E-bound (Upper_Arc (L~ (Cage (C,k))))) is V11() real ext-real Element of REAL
((W-bound (L~ (Cage (C,k)))) + (E-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2 is V11() real ext-real Element of COMPLEX
(UMP C) `2 is V11() real ext-real Element of REAL
Upper_Arc (L~ (Cage (C,1))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,1)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,1)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,1)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,1))))) + (W-bound (Upper_Arc (L~ (Cage (C,1))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,1))))) + (W-bound (Upper_Arc (L~ (Cage (C,1)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,1))))) + (W-bound (Upper_Arc (L~ (Cage (C,1)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,1))))) + (W-bound (Upper_Arc (L~ (Cage (C,1)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,1)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,1))))) + (W-bound (Upper_Arc (L~ (Cage (C,1)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,1)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,1))))) + (W-bound (Upper_Arc (L~ (Cage (C,1)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,1)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,1))))) + (W-bound (Upper_Arc (L~ (Cage (C,1)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,1))))) + (W-bound (Upper_Arc (L~ (Cage (C,1)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,1)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,1))))) + (W-bound (Upper_Arc (L~ (Cage (C,1)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,1))))) `2 is V11() real ext-real Element of REAL
LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound C)]|) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
(LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound C)]|)) /\ C is functional Element of K6( the carrier of (TOP-REAL 2))
{(UMP C)} is non empty functional Element of K6( the carrier of (TOP-REAL 2))
(UMP C) `1 is V11() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,k) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,k)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,k))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,k)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,k)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,k)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,k))))) + (W-bound (Upper_Arc (L~ (Cage (C,k)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,k))))) `2 is V11() real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
N-bound (L~ (Cage (C,k))) is V11() real ext-real Element of REAL
(UMP (Upper_Arc (L~ (Cage (C,k))))) `1 is V11() real ext-real Element of REAL
proj2 . (UMP (Upper_Arc (L~ (Cage (C,k))))) is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO)))]| `1 is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO)))]| `2 is V11() real ext-real Element of REAL
the carrier of (TopSpaceMetr (Euclid 2)) is set
K6( the carrier of (TopSpaceMetr (Euclid 2))) is set
dist_min ((LSeg (|[(((W-bound C) + (E-bound C)) / 2),(lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO)))]|,|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)),C) is V11() real ext-real Element of REAL
k is Element of K6( the carrier of (TopSpaceMetr (Euclid 2)))
R is Element of K6( the carrier of (TopSpaceMetr (Euclid 2)))
min_dist_min (k,R) is V11() real ext-real Element of REAL
((UMP C) `2) - (lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO))) is V11() real ext-real Element of REAL
(lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO))) + (((UMP C) `2) - (lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO)))) is V11() real ext-real Element of REAL
pp is V11() real ext-real set
S is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
proj2 . S is V11() real ext-real Element of REAL
S `2 is V11() real ext-real Element of REAL
pp is set
S is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
S `2 is V11() real ext-real Element of REAL
S `1 is V11() real ext-real Element of REAL
(dist_min ((LSeg (|[(((W-bound C) + (E-bound C)) / 2),(lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO)))]|,|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)),C)) / 2 is V11() real ext-real Element of COMPLEX
pp is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,pp) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,pp)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,pp)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,pp)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * (2,1))) is V11() real ext-real Element of REAL
Cage (C,pp) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,pp)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,pp))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,pp)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,pp)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,pp)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,pp))))) `2 is V11() real ext-real Element of REAL
N-bound (L~ (Cage (C,pp))) is V11() real ext-real Element of REAL
((dist_min ((LSeg (|[(((W-bound C) + (E-bound C)) / 2),(lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO)))]|,|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)),C)) / 2) + ((dist_min ((LSeg (|[(((W-bound C) + (E-bound C)) / 2),(lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO)))]|,|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)),C)) / 2) is V11() real ext-real Element of COMPLEX
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,pp)) * ((1 + 1),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * ((1 + 1),1))) is V11() real ext-real Element of REAL
(Gauge (C,pp)) * (1,(1 + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * (1,(1 + 1)))) is V11() real ext-real Element of REAL
(dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * ((1 + 1),1)))) + (dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * (1,(1 + 1))))) is V11() real ext-real Element of REAL
[1,(1 + 1)] is Element of K7(NAT,NAT)
{1,(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{1,(1 + 1)},{1}} is non empty set
Indices (Gauge (C,pp)) is set
S-bound C is V11() real ext-real Element of REAL
(N-bound C) - (S-bound C) is V11() real ext-real Element of REAL
2 |^ pp is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((N-bound C) - (S-bound C)) / (2 |^ pp) is V11() real ext-real Element of COMPLEX
[(1 + 1),1] is Element of K7(NAT,NAT)
{(1 + 1),1} is non empty V158() V159() V160() V161() V162() V163() set
{(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(1 + 1),1},{(1 + 1)}} is non empty set
(E-bound C) - (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) - (W-bound C)) / (2 |^ pp) is V11() real ext-real Element of COMPLEX
len (Cage (C,pp)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
LSeg ((Cage (C,pp)),n) is functional Element of K6( the carrier of (TOP-REAL 2))
(Cage (C,pp)) /. n is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Cage (C,pp)) /. (n + 1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[m,n] is Element of K7(NAT,NAT)
{m,n} is non empty V158() V159() V160() V161() V162() V163() set
{m} is non empty V158() V159() V160() V161() V162() V163() set
{{m,n},{m}} is non empty set
(Gauge (C,pp)) * (m,n) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Sm is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[m,Sm] is Element of K7(NAT,NAT)
{m,Sm} is non empty V158() V159() V160() V161() V162() V163() set
{m} is non empty V158() V159() V160() V161() V162() V163() set
{{m,Sm},{m}} is non empty set
(Gauge (C,pp)) * (m,Sm) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Sm + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
right_cell ((Cage (C,pp)),n,(Gauge (C,pp))) is functional Element of K6( the carrier of (TOP-REAL 2))
SSm is set
the carrier of (Euclid 2) is non empty set
r is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,pp))))) `1 is V11() real ext-real Element of REAL
s9 is Element of the carrier of (Euclid 2)
c9 is Element of the carrier of (Euclid 2)
dist (s9,c9) is V11() real ext-real Element of REAL
dist ((UMP (Upper_Arc (L~ (Cage (C,pp))))),r) is V11() real ext-real Element of REAL
left_cell ((Cage (C,pp)),n,(Gauge (C,pp))) is functional Element of K6( the carrier of (TOP-REAL 2))
(left_cell ((Cage (C,pp)),n,(Gauge (C,pp)))) /\ (right_cell ((Cage (C,pp)),n,(Gauge (C,pp)))) is functional Element of K6( the carrier of (TOP-REAL 2))
width (Gauge (C,pp)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Sm + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
m + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
len (Gauge (C,pp)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(m + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,pp)) * (m,(n + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,n)),((Gauge (C,pp)) * (m,(n + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,(n + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,n)) `2 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * (m,(n + 1))) `2) - (((Gauge (C,pp)) * (m,n)) `2) is V11() real ext-real Element of REAL
[(m + 1),n] is Element of K7(NAT,NAT)
{(m + 1),n} is non empty V158() V159() V160() V161() V162() V163() set
{(m + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(m + 1),n},{(m + 1)}} is non empty set
(Gauge (C,pp)) * ((m + 1),n) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,n)),((Gauge (C,pp)) * ((m + 1),n))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m + 1),n)) `1 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,n)) `1 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * ((m + 1),n)) `1) - (((Gauge (C,pp)) * (m,n)) `1) is V11() real ext-real Element of REAL
cell ((Gauge (C,pp)),m,n) is functional Element of K6( the carrier of (TOP-REAL 2))
r `1 is V11() real ext-real Element of REAL
r `2 is V11() real ext-real Element of REAL
((((Gauge (C,pp)) * ((m + 1),n)) `1) - (((Gauge (C,pp)) * (m,n)) `1)) + ((((Gauge (C,pp)) * (m,(n + 1))) `2) - (((Gauge (C,pp)) * (m,n)) `2)) is V11() real ext-real Element of REAL
n -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(n -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[m,(n -' 1)] is Element of K7(NAT,NAT)
{m,(n -' 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{m,(n -' 1)},{m}} is non empty set
(Gauge (C,pp)) * (m,(n -' 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,pp)) * (m,((n -' 1) + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,(n -' 1))),((Gauge (C,pp)) * (m,((n -' 1) + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,((n -' 1) + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,(n -' 1))) `2 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * (m,((n -' 1) + 1))) `2) - (((Gauge (C,pp)) * (m,(n -' 1))) `2) is V11() real ext-real Element of REAL
[(m + 1),(n -' 1)] is Element of K7(NAT,NAT)
{(m + 1),(n -' 1)} is non empty V158() V159() V160() V161() V162() V163() set
{(m + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(m + 1),(n -' 1)},{(m + 1)}} is non empty set
(Gauge (C,pp)) * ((m + 1),(n -' 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,(n -' 1))),((Gauge (C,pp)) * ((m + 1),(n -' 1)))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m + 1),(n -' 1))) `1 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,(n -' 1))) `1 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * ((m + 1),(n -' 1))) `1) - (((Gauge (C,pp)) * (m,(n -' 1))) `1) is V11() real ext-real Element of REAL
cell ((Gauge (C,pp)),m,(n -' 1)) is functional Element of K6( the carrier of (TOP-REAL 2))
r `1 is V11() real ext-real Element of REAL
r `2 is V11() real ext-real Element of REAL
((((Gauge (C,pp)) * ((m + 1),(n -' 1))) `1) - (((Gauge (C,pp)) * (m,(n -' 1))) `1)) + ((((Gauge (C,pp)) * (m,((n -' 1) + 1))) `2) - (((Gauge (C,pp)) * (m,(n -' 1))) `2)) is V11() real ext-real Element of REAL
(Gauge (C,pp)) * ((m + 1),Sm) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,Sm)),((Gauge (C,pp)) * ((m + 1),Sm))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m + 1),Sm)) `1 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,Sm)) `1 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * ((m + 1),Sm)) `1) - (((Gauge (C,pp)) * (m,Sm)) `1) is V11() real ext-real Element of REAL
[m,(Sm + 1)] is Element of K7(NAT,NAT)
{m,(Sm + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{m,(Sm + 1)},{m}} is non empty set
(Gauge (C,pp)) * (m,(Sm + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,Sm)),((Gauge (C,pp)) * (m,(Sm + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,(Sm + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,Sm)) `2 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * (m,(Sm + 1))) `2) - (((Gauge (C,pp)) * (m,Sm)) `2) is V11() real ext-real Element of REAL
cell ((Gauge (C,pp)),m,Sm) is functional Element of K6( the carrier of (TOP-REAL 2))
r `1 is V11() real ext-real Element of REAL
r `2 is V11() real ext-real Element of REAL
((((Gauge (C,pp)) * ((m + 1),Sm)) `1) - (((Gauge (C,pp)) * (m,Sm)) `1)) + ((((Gauge (C,pp)) * (m,(Sm + 1))) `2) - (((Gauge (C,pp)) * (m,Sm)) `2)) is V11() real ext-real Element of REAL
m -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[(m -' 1),Sm] is Element of K7(NAT,NAT)
{(m -' 1),Sm} is non empty V158() V159() V160() V161() V162() V163() set
{(m -' 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(m -' 1),Sm},{(m -' 1)}} is non empty set
[(m -' 1),(Sm + 1)] is Element of K7(NAT,NAT)
{(m -' 1),(Sm + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(m -' 1),(Sm + 1)},{(m -' 1)}} is non empty set
(Gauge (C,pp)) * ((m -' 1),Sm) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,pp)) * ((m -' 1),(Sm + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * ((m -' 1),Sm)),((Gauge (C,pp)) * ((m -' 1),(Sm + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m -' 1),(Sm + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m -' 1),Sm)) `2 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * ((m -' 1),(Sm + 1))) `2) - (((Gauge (C,pp)) * ((m -' 1),Sm)) `2) is V11() real ext-real Element of REAL
(m -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[((m -' 1) + 1),Sm] is Element of K7(NAT,NAT)
{((m -' 1) + 1),Sm} is non empty V158() V159() V160() V161() V162() V163() set
{((m -' 1) + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{((m -' 1) + 1),Sm},{((m -' 1) + 1)}} is non empty set
(Gauge (C,pp)) * (((m -' 1) + 1),Sm) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * ((m -' 1),Sm)),((Gauge (C,pp)) * (((m -' 1) + 1),Sm))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (((m -' 1) + 1),Sm)) `1 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m -' 1),Sm)) `1 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * (((m -' 1) + 1),Sm)) `1) - (((Gauge (C,pp)) * ((m -' 1),Sm)) `1) is V11() real ext-real Element of REAL
cell ((Gauge (C,pp)),(m -' 1),Sm) is functional Element of K6( the carrier of (TOP-REAL 2))
r `1 is V11() real ext-real Element of REAL
r `2 is V11() real ext-real Element of REAL
((((Gauge (C,pp)) * (((m -' 1) + 1),Sm)) `1) - (((Gauge (C,pp)) * ((m -' 1),Sm)) `1)) + ((((Gauge (C,pp)) * ((m -' 1),(Sm + 1))) `2) - (((Gauge (C,pp)) * ((m -' 1),Sm)) `2)) is V11() real ext-real Element of REAL
K7(NAT, the carrier of (TOP-REAL 2)) is set
K6(K7(NAT, the carrier of (TOP-REAL 2))) is set
Upper_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
{ ((UMP (Upper_Arc (L~ (Cage (C,b1))))) `2) where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT : not b1 <= 0 } is set
R is set
pp is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,pp) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,pp)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,pp))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,pp)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,pp)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,pp)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,pp))))) + (W-bound (Upper_Arc (L~ (Cage (C,pp)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,pp))))) `2 is V11() real ext-real Element of REAL
REAL 2 is non empty FinSequence-membered M9( REAL )
SS is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,SS) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,SS)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,SS))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,SS)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,SS)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,SS)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,SS))))) + (W-bound (Upper_Arc (L~ (Cage (C,SS))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,SS))))) + (W-bound (Upper_Arc (L~ (Cage (C,SS)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,SS))))) + (W-bound (Upper_Arc (L~ (Cage (C,SS)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,SS))))) + (W-bound (Upper_Arc (L~ (Cage (C,SS)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,SS)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,SS))))) + (W-bound (Upper_Arc (L~ (Cage (C,SS)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,SS)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,SS))))) + (W-bound (Upper_Arc (L~ (Cage (C,SS)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,SS)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,SS))))) + (W-bound (Upper_Arc (L~ (Cage (C,SS)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,SS))))) + (W-bound (Upper_Arc (L~ (Cage (C,SS)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,SS)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,SS))))) + (W-bound (Upper_Arc (L~ (Cage (C,SS)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
K7(NAT,(REAL 2)) is set
K6(K7(NAT,(REAL 2))) is set
SS is V19() V22( NAT ) V23( REAL 2) Function-like V46( NAT , REAL 2) Element of K6(K7(NAT,(REAL 2)))
r is V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like V46( NAT , the carrier of (TOP-REAL 2)) Element of K6(K7(NAT, the carrier of (TOP-REAL 2)))
lim r is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
R is V158() V159() V160() Element of K6(REAL)
n is ext-real set
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,m) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,m)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,m))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,m)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,m)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,m)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,m))))) `2 is V11() real ext-real Element of REAL
pp is V41(2) Element of REAL 2
n is V11() real ext-real set
(lower_bound (proj2 .: ((LSeg ((UMP C),|[(((W-bound C) + (E-bound C)) / 2),(N-bound (L~ (Cage (C,1))))]|)) /\ OO))) + n is V11() real ext-real Element of REAL
m is V11() real ext-real set
n is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
proj2 . n is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,m) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,m)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,m))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,m)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,m)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,m)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,m))))) `2 is V11() real ext-real Element of REAL
n is V11() real ext-real Element of REAL
lower_bound R is V11() real ext-real Element of REAL
(lower_bound R) + n is V11() real ext-real Element of REAL
m is V11() real ext-real set
((lower_bound R) + n) - (lower_bound R) is V11() real ext-real Element of REAL
m - (lower_bound R) is V11() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,n) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,n)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,n))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,n)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,n)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,n)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,n))))) `2 is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
SS . m is V41(2) Element of REAL 2
(SS . m) - pp is V19() V22( NAT ) V23( REAL ) Function-like V41(2) FinSequence-like V148() V149() V150() M10( REAL , REAL 2)
|.((SS . m) - pp).| is V11() real ext-real non negative Element of REAL
Cage (C,m) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,m)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,m))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,m)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,m)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,m)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,m))))) + (W-bound (Upper_Arc (L~ (Cage (C,m)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,m))))) `2 is V11() real ext-real Element of REAL
Sm is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Sm `2 is V11() real ext-real Element of REAL
(Sm `2) - ((UMP C) `2) is V11() real ext-real Element of REAL
m - ((UMP C) `2) is V11() real ext-real Element of REAL
SSm is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
S is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
SSm - S is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
K438(SSm,S) is V19() V22( NAT ) V23( REAL ) Function-like FinSequence-like V148() V149() V150() FinSequence of REAL
Sm `1 is V11() real ext-real Element of REAL
(Sm `1) - ((UMP C) `1) is V11() real ext-real Element of REAL
abs ((Sm `1) - ((UMP C) `1)) is V11() real ext-real Element of REAL
abs ((Sm `2) - ((UMP C) `2)) is V11() real ext-real Element of REAL
0 + (abs ((Sm `2) - ((UMP C) `2))) is V11() real ext-real Element of REAL
r is V11() real ext-real set
s9 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,s9) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,s9)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,s9))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP (Upper_Arc (L~ (Cage (C,s9)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,s9)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,s9)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,s9))))) + (W-bound (Upper_Arc (L~ (Cage (C,s9))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,s9))))) + (W-bound (Upper_Arc (L~ (Cage (C,s9)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,s9))))) + (W-bound (Upper_Arc (L~ (Cage (C,s9)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,s9))))) + (W-bound (Upper_Arc (L~ (Cage (C,s9)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,s9)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,s9))))) + (W-bound (Upper_Arc (L~ (Cage (C,s9)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,s9)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,s9))))) + (W-bound (Upper_Arc (L~ (Cage (C,s9)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,s9)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,s9))))) + (W-bound (Upper_Arc (L~ (Cage (C,s9)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,s9))))) + (W-bound (Upper_Arc (L~ (Cage (C,s9)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,s9)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,s9))))) + (W-bound (Upper_Arc (L~ (Cage (C,s9)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(UMP (Upper_Arc (L~ (Cage (C,s9))))) `2 is V11() real ext-real Element of REAL
n is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
r . n is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(r . n) - (UMP C) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
K438((r . n),(UMP C)) is V19() V22( NAT ) V23( REAL ) Function-like FinSequence-like V148() V149() V150() FinSequence of REAL
|.((r . n) - (UMP C)).| is V11() real ext-real non negative Element of REAL
SS . n is V41(2) Element of REAL 2
(SS . n) - pp is V19() V22( NAT ) V23( REAL ) Function-like V41(2) FinSequence-like V148() V149() V150() M10( REAL , REAL 2)
|.((SS . n) - pp).| is V11() real ext-real non negative Element of REAL
r . n is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(r . n) - (UMP C) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
K438((r . n),(UMP C)) is V19() V22( NAT ) V23( REAL ) Function-like FinSequence-like V148() V149() V150() FinSequence of REAL
|.((r . n) - (UMP C)).| is V11() real ext-real non negative Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Upper_Appr C) . n is functional Element of K636( the carrier of (TOP-REAL 2))
Cage (C,n) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,n)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,n))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
SS . n is V41(2) Element of REAL 2
UMP (Upper_Arc (L~ (Cage (C,n)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Upper_Arc (L~ (Cage (C,n)))) is V11() real ext-real Element of REAL
W-bound (Upper_Arc (L~ (Cage (C,n)))) is V11() real ext-real Element of REAL
(E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n))))) is V11() real ext-real Element of REAL
((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2 } is set
(Upper_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Upper_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2),(upper_bound (proj2 .: ((Upper_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Upper_Arc (L~ (Cage (C,n))))) + (W-bound (Upper_Arc (L~ (Cage (C,n)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
r . n is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
n is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
r . n is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(r . n) - (UMP C) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
K438((r . n),(UMP C)) is V19() V22( NAT ) V23( REAL ) Function-like FinSequence-like V148() V149() V150() FinSequence of REAL
|.((r . n) - (UMP C)).| is V11() real ext-real non negative Element of REAL
SS . n is V41(2) Element of REAL 2
(SS . n) - pp is V19() V22( NAT ) V23( REAL ) Function-like V41(2) FinSequence-like V148() V149() V150() M10( REAL , REAL 2)
|.((SS . n) - pp).| is V11() real ext-real non negative Element of REAL
Lim_inf (Upper_Appr C) is functional Element of K6( the carrier of (TOP-REAL 2))
k is V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like V46( NAT , the carrier of (TOP-REAL 2)) Element of K6(K7(NAT, the carrier of (TOP-REAL 2)))
lim k is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
LMP C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound C) + (W-bound C)) / 2),(lower_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
South_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
(W-bound C) + (E-bound C) is V11() real ext-real Element of REAL
((W-bound C) + (E-bound C)) / 2 is V11() real ext-real Element of COMPLEX
{ (LMP (Lower_Arc (L~ (Cage (C,b1))))) where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT : not b1 <= 0 } is set
Cage (C,1) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,1)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
S-bound (L~ (Cage (C,1))) is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
S-bound C is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(S-bound C)]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
|[(((W-bound C) + (E-bound C)) / 2),(S-bound C)]| `1 is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]| `1 is V11() real ext-real Element of REAL
OO is set
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,r) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,r)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,r))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,r)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,r)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,r)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
OO is functional Element of K6( the carrier of (TOP-REAL 2))
(LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO) is V158() V159() V160() Element of K6(REAL)
upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO)) is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO)))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
LSeg (|[(((W-bound C) + (E-bound C)) / 2),(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO)))]|,|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
|[(((W-bound C) + (E-bound C)) / 2),(S-bound C)]| `2 is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]| `2 is V11() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,k) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,k)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,k))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,k)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,k)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,k)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,k))))) `1 is V11() real ext-real Element of REAL
W-bound (L~ (Cage (C,k))) is V11() real ext-real Element of REAL
E-bound (L~ (Cage (C,k))) is V11() real ext-real Element of REAL
(W-bound (L~ (Cage (C,k)))) + (E-bound (L~ (Cage (C,k)))) is V11() real ext-real Element of REAL
(W-bound (Lower_Arc (L~ (Cage (C,k))))) + (E-bound (Lower_Arc (L~ (Cage (C,k))))) is V11() real ext-real Element of REAL
((W-bound (Lower_Arc (L~ (Cage (C,k))))) + (E-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2 is V11() real ext-real Element of COMPLEX
(W-bound (L~ (Cage (C,k)))) + (E-bound (Lower_Arc (L~ (Cage (C,k))))) is V11() real ext-real Element of REAL
((W-bound (L~ (Cage (C,k)))) + (E-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2 is V11() real ext-real Element of COMPLEX
Lower_Arc (L~ (Cage (C,1))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,1)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,1)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,1)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,1))))) + (W-bound (Lower_Arc (L~ (Cage (C,1))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,1))))) + (W-bound (Lower_Arc (L~ (Cage (C,1)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,1))))) + (W-bound (Lower_Arc (L~ (Cage (C,1)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,1))))) + (W-bound (Lower_Arc (L~ (Cage (C,1)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,1)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,1))))) + (W-bound (Lower_Arc (L~ (Cage (C,1)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,1)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,1))))) + (W-bound (Lower_Arc (L~ (Cage (C,1)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,1)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,1))))) + (W-bound (Lower_Arc (L~ (Cage (C,1)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,1))))) + (W-bound (Lower_Arc (L~ (Cage (C,1)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,1)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,1))))) + (W-bound (Lower_Arc (L~ (Cage (C,1)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,1))))) `2 is V11() real ext-real Element of REAL
(LMP C) `2 is V11() real ext-real Element of REAL
LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound C)]|) is non empty functional closed closed connected bounded bounded compact compact convex Element of K6( the carrier of (TOP-REAL 2))
(LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound C)]|)) /\ C is functional Element of K6( the carrier of (TOP-REAL 2))
{(LMP C)} is non empty functional Element of K6( the carrier of (TOP-REAL 2))
(LMP C) `1 is V11() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,k) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,k)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,k))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,k)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,k)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,k)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,k)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,k))))) + (W-bound (Lower_Arc (L~ (Cage (C,k)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,k))))) `2 is V11() real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
S-bound (L~ (Cage (C,k))) is V11() real ext-real Element of REAL
(LMP (Lower_Arc (L~ (Cage (C,k))))) `1 is V11() real ext-real Element of REAL
proj2 . (LMP (Lower_Arc (L~ (Cage (C,k))))) is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO)))]| `1 is V11() real ext-real Element of REAL
|[(((W-bound C) + (E-bound C)) / 2),(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO)))]| `2 is V11() real ext-real Element of REAL
the carrier of (TopSpaceMetr (Euclid 2)) is set
K6( the carrier of (TopSpaceMetr (Euclid 2))) is set
dist_min ((LSeg (|[(((W-bound C) + (E-bound C)) / 2),(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO)))]|,|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)),C) is V11() real ext-real Element of REAL
k is Element of K6( the carrier of (TopSpaceMetr (Euclid 2)))
R is Element of K6( the carrier of (TopSpaceMetr (Euclid 2)))
min_dist_min (k,R) is V11() real ext-real Element of REAL
(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO))) - ((LMP C) `2) is V11() real ext-real Element of REAL
(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO))) - ((upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO))) - ((LMP C) `2)) is V11() real ext-real Element of REAL
pp is V11() real ext-real set
S is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
proj2 . S is V11() real ext-real Element of REAL
S `2 is V11() real ext-real Element of REAL
pp is set
S is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
S `2 is V11() real ext-real Element of REAL
S `1 is V11() real ext-real Element of REAL
(dist_min ((LSeg (|[(((W-bound C) + (E-bound C)) / 2),(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO)))]|,|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)),C)) / 2 is V11() real ext-real Element of COMPLEX
pp is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Gauge (C,pp) is V19() non empty-yielding V22( NAT ) V23(K285( the carrier of (TOP-REAL 2))) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of K285( the carrier of (TOP-REAL 2))
(Gauge (C,pp)) * (1,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,pp)) * (1,2) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * (1,2))) is V11() real ext-real Element of REAL
(Gauge (C,pp)) * (2,1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * (2,1))) is V11() real ext-real Element of REAL
Cage (C,pp) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,pp)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,pp))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,pp)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,pp)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,pp)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
S-bound (L~ (Cage (C,pp))) is V11() real ext-real Element of REAL
(LMP (Lower_Arc (L~ (Cage (C,pp))))) `2 is V11() real ext-real Element of REAL
((dist_min ((LSeg (|[(((W-bound C) + (E-bound C)) / 2),(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO)))]|,|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)),C)) / 2) + ((dist_min ((LSeg (|[(((W-bound C) + (E-bound C)) / 2),(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO)))]|,|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)),C)) / 2) is V11() real ext-real Element of COMPLEX
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,pp)) * ((1 + 1),1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * ((1 + 1),1))) is V11() real ext-real Element of REAL
(Gauge (C,pp)) * (1,(1 + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * (1,(1 + 1)))) is V11() real ext-real Element of REAL
(dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * ((1 + 1),1)))) + (dist (((Gauge (C,pp)) * (1,1)),((Gauge (C,pp)) * (1,(1 + 1))))) is V11() real ext-real Element of REAL
[1,(1 + 1)] is Element of K7(NAT,NAT)
{1,(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{1,(1 + 1)},{1}} is non empty set
Indices (Gauge (C,pp)) is set
N-bound C is V11() real ext-real Element of REAL
(N-bound C) - (S-bound C) is V11() real ext-real Element of REAL
2 |^ pp is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
((N-bound C) - (S-bound C)) / (2 |^ pp) is V11() real ext-real Element of COMPLEX
[(1 + 1),1] is Element of K7(NAT,NAT)
{(1 + 1),1} is non empty V158() V159() V160() V161() V162() V163() set
{(1 + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(1 + 1),1},{(1 + 1)}} is non empty set
(E-bound C) - (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) - (W-bound C)) / (2 |^ pp) is V11() real ext-real Element of COMPLEX
len (Cage (C,pp)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
LSeg ((Cage (C,pp)),n) is functional Element of K6( the carrier of (TOP-REAL 2))
(Cage (C,pp)) /. n is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Cage (C,pp)) /. (n + 1) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[m,n] is Element of K7(NAT,NAT)
{m,n} is non empty V158() V159() V160() V161() V162() V163() set
{m} is non empty V158() V159() V160() V161() V162() V163() set
{{m,n},{m}} is non empty set
(Gauge (C,pp)) * (m,n) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Sm is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[m,Sm] is Element of K7(NAT,NAT)
{m,Sm} is non empty V158() V159() V160() V161() V162() V163() set
{m} is non empty V158() V159() V160() V161() V162() V163() set
{{m,Sm},{m}} is non empty set
(Gauge (C,pp)) * (m,Sm) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Sm + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
right_cell ((Cage (C,pp)),n,(Gauge (C,pp))) is functional Element of K6( the carrier of (TOP-REAL 2))
SSm is set
the carrier of (Euclid 2) is non empty set
r is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,pp))))) `1 is V11() real ext-real Element of REAL
s9 is Element of the carrier of (Euclid 2)
c9 is Element of the carrier of (Euclid 2)
dist (s9,c9) is V11() real ext-real Element of REAL
dist ((LMP (Lower_Arc (L~ (Cage (C,pp))))),r) is V11() real ext-real Element of REAL
left_cell ((Cage (C,pp)),n,(Gauge (C,pp))) is functional Element of K6( the carrier of (TOP-REAL 2))
(left_cell ((Cage (C,pp)),n,(Gauge (C,pp)))) /\ (right_cell ((Cage (C,pp)),n,(Gauge (C,pp)))) is functional Element of K6( the carrier of (TOP-REAL 2))
width (Gauge (C,pp)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Sm + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
m + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
len (Gauge (C,pp)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(m + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Gauge (C,pp)) * (m,(n + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,n)),((Gauge (C,pp)) * (m,(n + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,(n + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,n)) `2 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * (m,(n + 1))) `2) - (((Gauge (C,pp)) * (m,n)) `2) is V11() real ext-real Element of REAL
[(m + 1),n] is Element of K7(NAT,NAT)
{(m + 1),n} is non empty V158() V159() V160() V161() V162() V163() set
{(m + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(m + 1),n},{(m + 1)}} is non empty set
(Gauge (C,pp)) * ((m + 1),n) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,n)),((Gauge (C,pp)) * ((m + 1),n))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m + 1),n)) `1 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,n)) `1 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * ((m + 1),n)) `1) - (((Gauge (C,pp)) * (m,n)) `1) is V11() real ext-real Element of REAL
cell ((Gauge (C,pp)),m,n) is functional Element of K6( the carrier of (TOP-REAL 2))
r `1 is V11() real ext-real Element of REAL
r `2 is V11() real ext-real Element of REAL
((((Gauge (C,pp)) * ((m + 1),n)) `1) - (((Gauge (C,pp)) * (m,n)) `1)) + ((((Gauge (C,pp)) * (m,(n + 1))) `2) - (((Gauge (C,pp)) * (m,n)) `2)) is V11() real ext-real Element of REAL
n -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(n -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[m,(n -' 1)] is Element of K7(NAT,NAT)
{m,(n -' 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{m,(n -' 1)},{m}} is non empty set
(Gauge (C,pp)) * (m,(n -' 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,pp)) * (m,((n -' 1) + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,(n -' 1))),((Gauge (C,pp)) * (m,((n -' 1) + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,((n -' 1) + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,(n -' 1))) `2 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * (m,((n -' 1) + 1))) `2) - (((Gauge (C,pp)) * (m,(n -' 1))) `2) is V11() real ext-real Element of REAL
[(m + 1),(n -' 1)] is Element of K7(NAT,NAT)
{(m + 1),(n -' 1)} is non empty V158() V159() V160() V161() V162() V163() set
{(m + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(m + 1),(n -' 1)},{(m + 1)}} is non empty set
(Gauge (C,pp)) * ((m + 1),(n -' 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,(n -' 1))),((Gauge (C,pp)) * ((m + 1),(n -' 1)))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m + 1),(n -' 1))) `1 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,(n -' 1))) `1 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * ((m + 1),(n -' 1))) `1) - (((Gauge (C,pp)) * (m,(n -' 1))) `1) is V11() real ext-real Element of REAL
cell ((Gauge (C,pp)),m,(n -' 1)) is functional Element of K6( the carrier of (TOP-REAL 2))
r `1 is V11() real ext-real Element of REAL
r `2 is V11() real ext-real Element of REAL
((((Gauge (C,pp)) * ((m + 1),(n -' 1))) `1) - (((Gauge (C,pp)) * (m,(n -' 1))) `1)) + ((((Gauge (C,pp)) * (m,((n -' 1) + 1))) `2) - (((Gauge (C,pp)) * (m,(n -' 1))) `2)) is V11() real ext-real Element of REAL
(Gauge (C,pp)) * ((m + 1),Sm) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,Sm)),((Gauge (C,pp)) * ((m + 1),Sm))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m + 1),Sm)) `1 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,Sm)) `1 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * ((m + 1),Sm)) `1) - (((Gauge (C,pp)) * (m,Sm)) `1) is V11() real ext-real Element of REAL
[m,(Sm + 1)] is Element of K7(NAT,NAT)
{m,(Sm + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{m,(Sm + 1)},{m}} is non empty set
(Gauge (C,pp)) * (m,(Sm + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * (m,Sm)),((Gauge (C,pp)) * (m,(Sm + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,(Sm + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (m,Sm)) `2 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * (m,(Sm + 1))) `2) - (((Gauge (C,pp)) * (m,Sm)) `2) is V11() real ext-real Element of REAL
cell ((Gauge (C,pp)),m,Sm) is functional Element of K6( the carrier of (TOP-REAL 2))
r `1 is V11() real ext-real Element of REAL
r `2 is V11() real ext-real Element of REAL
((((Gauge (C,pp)) * ((m + 1),Sm)) `1) - (((Gauge (C,pp)) * (m,Sm)) `1)) + ((((Gauge (C,pp)) * (m,(Sm + 1))) `2) - (((Gauge (C,pp)) * (m,Sm)) `2)) is V11() real ext-real Element of REAL
m -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[(m -' 1),Sm] is Element of K7(NAT,NAT)
{(m -' 1),Sm} is non empty V158() V159() V160() V161() V162() V163() set
{(m -' 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(m -' 1),Sm},{(m -' 1)}} is non empty set
[(m -' 1),(Sm + 1)] is Element of K7(NAT,NAT)
{(m -' 1),(Sm + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{(m -' 1),(Sm + 1)},{(m -' 1)}} is non empty set
(Gauge (C,pp)) * ((m -' 1),Sm) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(Gauge (C,pp)) * ((m -' 1),(Sm + 1)) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * ((m -' 1),Sm)),((Gauge (C,pp)) * ((m -' 1),(Sm + 1)))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m -' 1),(Sm + 1))) `2 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m -' 1),Sm)) `2 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * ((m -' 1),(Sm + 1))) `2) - (((Gauge (C,pp)) * ((m -' 1),Sm)) `2) is V11() real ext-real Element of REAL
(m -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
[((m -' 1) + 1),Sm] is Element of K7(NAT,NAT)
{((m -' 1) + 1),Sm} is non empty V158() V159() V160() V161() V162() V163() set
{((m -' 1) + 1)} is non empty V158() V159() V160() V161() V162() V163() set
{{((m -' 1) + 1),Sm},{((m -' 1) + 1)}} is non empty set
(Gauge (C,pp)) * (((m -' 1) + 1),Sm) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
dist (((Gauge (C,pp)) * ((m -' 1),Sm)),((Gauge (C,pp)) * (((m -' 1) + 1),Sm))) is V11() real ext-real Element of REAL
((Gauge (C,pp)) * (((m -' 1) + 1),Sm)) `1 is V11() real ext-real Element of REAL
((Gauge (C,pp)) * ((m -' 1),Sm)) `1 is V11() real ext-real Element of REAL
(((Gauge (C,pp)) * (((m -' 1) + 1),Sm)) `1) - (((Gauge (C,pp)) * ((m -' 1),Sm)) `1) is V11() real ext-real Element of REAL
cell ((Gauge (C,pp)),(m -' 1),Sm) is functional Element of K6( the carrier of (TOP-REAL 2))
r `1 is V11() real ext-real Element of REAL
r `2 is V11() real ext-real Element of REAL
((((Gauge (C,pp)) * (((m -' 1) + 1),Sm)) `1) - (((Gauge (C,pp)) * ((m -' 1),Sm)) `1)) + ((((Gauge (C,pp)) * ((m -' 1),(Sm + 1))) `2) - (((Gauge (C,pp)) * ((m -' 1),Sm)) `2)) is V11() real ext-real Element of REAL
K7(NAT, the carrier of (TOP-REAL 2)) is set
K6(K7(NAT, the carrier of (TOP-REAL 2))) is set
Lower_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
{ ((LMP (Lower_Arc (L~ (Cage (C,b1))))) `2) where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT : not b1 <= 0 } is set
R is set
pp is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,pp) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,pp)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,pp))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,pp)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,pp)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,pp)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,pp)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,pp))))) + (W-bound (Lower_Arc (L~ (Cage (C,pp)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,pp))))) `2 is V11() real ext-real Element of REAL
REAL 2 is non empty FinSequence-membered M9( REAL )
S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,S) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,S)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,S))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,S)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,S)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,S)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,S))))) + (W-bound (Lower_Arc (L~ (Cage (C,S))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,S))))) + (W-bound (Lower_Arc (L~ (Cage (C,S)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,S))))) + (W-bound (Lower_Arc (L~ (Cage (C,S)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,S))))) + (W-bound (Lower_Arc (L~ (Cage (C,S)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,S)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,S))))) + (W-bound (Lower_Arc (L~ (Cage (C,S)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,S)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,S))))) + (W-bound (Lower_Arc (L~ (Cage (C,S)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,S)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,S))))) + (W-bound (Lower_Arc (L~ (Cage (C,S)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,S))))) + (W-bound (Lower_Arc (L~ (Cage (C,S)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,S)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,S))))) + (W-bound (Lower_Arc (L~ (Cage (C,S)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
K7(NAT,(REAL 2)) is set
K6(K7(NAT,(REAL 2))) is set
S is V19() V22( NAT ) V23( REAL 2) Function-like V46( NAT , REAL 2) Element of K6(K7(NAT,(REAL 2)))
SS is V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like V46( NAT , the carrier of (TOP-REAL 2)) Element of K6(K7(NAT, the carrier of (TOP-REAL 2)))
lim SS is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
R is V158() V159() V160() Element of K6(REAL)
r is ext-real set
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,n) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,n)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,n))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,n)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,n)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,n)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,n))))) `2 is V11() real ext-real Element of REAL
pp is V41(2) Element of REAL 2
r is V11() real ext-real set
(upper_bound (proj2 .: ((LSeg ((LMP C),|[(((W-bound C) + (E-bound C)) / 2),(S-bound (L~ (Cage (C,1))))]|)) /\ OO))) - r is V11() real ext-real Element of REAL
n is V11() real ext-real set
m is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
proj2 . m is V11() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,n) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,n)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,n))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,n)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,n)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,n)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,n))))) `2 is V11() real ext-real Element of REAL
n is V11() real ext-real Element of REAL
upper_bound R is V11() real ext-real Element of REAL
(upper_bound R) - n is V11() real ext-real Element of REAL
m is V11() real ext-real set
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,n) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,n)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,n))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,n)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,n)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,n)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,n)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,n))))) + (W-bound (Lower_Arc (L~ (Cage (C,n)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,n))))) `2 is V11() real ext-real Element of REAL
m + n is V11() real ext-real Element of REAL
((upper_bound R) - n) + n is V11() real ext-real Element of REAL
(m + n) - m is V11() real ext-real Element of REAL
(upper_bound R) - m is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
S . m is V41(2) Element of REAL 2
(S . m) - pp is V19() V22( NAT ) V23( REAL ) Function-like V41(2) FinSequence-like V148() V149() V150() M10( REAL , REAL 2)
|.((S . m) - pp).| is V11() real ext-real non negative Element of REAL
Cage (C,m) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,m)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,m))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,m)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,m)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,m)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,m))))) + (W-bound (Lower_Arc (L~ (Cage (C,m))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,m))))) + (W-bound (Lower_Arc (L~ (Cage (C,m)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,m))))) + (W-bound (Lower_Arc (L~ (Cage (C,m)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,m))))) + (W-bound (Lower_Arc (L~ (Cage (C,m)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,m))))) + (W-bound (Lower_Arc (L~ (Cage (C,m)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,m))))) + (W-bound (Lower_Arc (L~ (Cage (C,m)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,m))))) + (W-bound (Lower_Arc (L~ (Cage (C,m)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,m))))) + (W-bound (Lower_Arc (L~ (Cage (C,m)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,m)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,m))))) + (W-bound (Lower_Arc (L~ (Cage (C,m)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,m))))) `2 is V11() real ext-real Element of REAL
Sm is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Sm `2 is V11() real ext-real Element of REAL
((LMP C) `2) - (Sm `2) is V11() real ext-real Element of REAL
((LMP C) `2) - m is V11() real ext-real Element of REAL
SSm is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
r is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
SSm - r is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
K438(SSm,r) is V19() V22( NAT ) V23( REAL ) Function-like FinSequence-like V148() V149() V150() FinSequence of REAL
Sm `1 is V11() real ext-real Element of REAL
(Sm `1) - ((LMP C) `1) is V11() real ext-real Element of REAL
abs ((Sm `1) - ((LMP C) `1)) is V11() real ext-real Element of REAL
(Sm `2) - ((LMP C) `2) is V11() real ext-real Element of REAL
abs ((Sm `2) - ((LMP C) `2)) is V11() real ext-real Element of REAL
0 + (abs ((Sm `2) - ((LMP C) `2))) is V11() real ext-real Element of REAL
- ((Sm `2) - ((LMP C) `2)) is V11() real ext-real Element of REAL
r is V11() real ext-real set
s9 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,s9) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,s9)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,s9))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
LMP (Lower_Arc (L~ (Cage (C,s9)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,s9)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,s9)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,s9))))) + (W-bound (Lower_Arc (L~ (Cage (C,s9))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,s9))))) + (W-bound (Lower_Arc (L~ (Cage (C,s9)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,s9))))) + (W-bound (Lower_Arc (L~ (Cage (C,s9)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,s9))))) + (W-bound (Lower_Arc (L~ (Cage (C,s9)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,s9)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,s9))))) + (W-bound (Lower_Arc (L~ (Cage (C,s9)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,s9)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,s9))))) + (W-bound (Lower_Arc (L~ (Cage (C,s9)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,s9)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,s9))))) + (W-bound (Lower_Arc (L~ (Cage (C,s9)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,s9))))) + (W-bound (Lower_Arc (L~ (Cage (C,s9)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,s9)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,s9))))) + (W-bound (Lower_Arc (L~ (Cage (C,s9)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(LMP (Lower_Arc (L~ (Cage (C,s9))))) `2 is V11() real ext-real Element of REAL
r is V11() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
SS . m is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(SS . m) - (LMP C) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
K438((SS . m),(LMP C)) is V19() V22( NAT ) V23( REAL ) Function-like FinSequence-like V148() V149() V150() FinSequence of REAL
|.((SS . m) - (LMP C)).| is V11() real ext-real non negative Element of REAL
S . m is V41(2) Element of REAL 2
(S . m) - pp is V19() V22( NAT ) V23( REAL ) Function-like V41(2) FinSequence-like V148() V149() V150() M10( REAL , REAL 2)
|.((S . m) - pp).| is V11() real ext-real non negative Element of REAL
SS . m is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(SS . m) - (LMP C) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
K438((SS . m),(LMP C)) is V19() V22( NAT ) V23( REAL ) Function-like FinSequence-like V148() V149() V150() FinSequence of REAL
|.((SS . m) - (LMP C)).| is V11() real ext-real non negative Element of REAL
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Lower_Appr C) . r is functional Element of K636( the carrier of (TOP-REAL 2))
Cage (C,r) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,r)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,r))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
S . r is V41(2) Element of REAL 2
LMP (Lower_Arc (L~ (Cage (C,r)))) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound (Lower_Arc (L~ (Cage (C,r)))) is V11() real ext-real Element of REAL
W-bound (Lower_Arc (L~ (Cage (C,r)))) is V11() real ext-real Element of REAL
(E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r))))) is V11() real ext-real Element of REAL
((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2 } is set
(Lower_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: ((Lower_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2),(lower_bound (proj2 .: ((Lower_Arc (L~ (Cage (C,r)))) /\ (Vertical_Line (((E-bound (Lower_Arc (L~ (Cage (C,r))))) + (W-bound (Lower_Arc (L~ (Cage (C,r)))))) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
SS . r is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
r is V11() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
SS . m is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
(SS . m) - (LMP C) is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
K438((SS . m),(LMP C)) is V19() V22( NAT ) V23( REAL ) Function-like FinSequence-like V148() V149() V150() FinSequence of REAL
|.((SS . m) - (LMP C)).| is V11() real ext-real non negative Element of REAL
S . m is V41(2) Element of REAL 2
(S . m) - pp is V19() V22( NAT ) V23( REAL ) Function-like V41(2) FinSequence-like V148() V149() V150() M10( REAL , REAL 2)
|.((S . m) - pp).| is V11() real ext-real non negative Element of REAL
Lim_inf (Lower_Appr C) is functional Element of K6( the carrier of (TOP-REAL 2))
k is V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like V46( NAT , the carrier of (TOP-REAL 2)) Element of K6(K7(NAT, the carrier of (TOP-REAL 2)))
lim k is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
North_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
Upper_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
Lim_inf (Upper_Appr C) is functional Element of K6( the carrier of (TOP-REAL 2))
the carrier of (TopSpaceMetr (Euclid 2)) is set
K6( the carrier of (TopSpaceMetr (Euclid 2))) is set
BDD C is functional open Element of K6( the carrier of (TOP-REAL 2))
x is set
e is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C ` is functional Element of K6( the carrier of (TOP-REAL 2))
UBD C is non empty functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
(BDD C) \/ (UBD C) is non empty functional Element of K6( the carrier of (TOP-REAL 2))
the carrier of (Euclid 2) is non empty set
x is Element of K6( the carrier of (TopSpaceMetr (Euclid 2)))
A is Element of the carrier of (Euclid 2)
m is V11() real ext-real set
Ball (A,m) is Element of K6( the carrier of (Euclid 2))
K6( the carrier of (Euclid 2)) is set
OO is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
OO + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,(OO + 1)) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,(OO + 1))) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,(OO + 1)))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
(Upper_Appr C) . (OO + 1) is functional Element of K636( the carrier of (TOP-REAL 2))
OO + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
UBD-Family C is non empty Element of K6(K6( the carrier of (TOP-REAL 2)))
union (UBD-Family C) is set
m is set
{ (UBD (L~ (Cage (C,b1)))) where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT : verum } is set
OO is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,OO) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,OO)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UBD (L~ (Cage (C,OO))) is non empty functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
LeftComp (Cage (C,OO)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
r is Element of K6( the carrier of (TopSpaceMetr (Euclid 2)))
A is Element of the carrier of (Euclid 2)
k is V11() real ext-real set
Ball (A,k) is Element of K6( the carrier of (Euclid 2))
K6( the carrier of (Euclid 2)) is set
k is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Upper_Appr C) . OO is functional Element of K636( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,OO))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
Cage (C,k) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
LeftComp (Cage (C,k)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
k + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,(k + 1)) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
LeftComp (Cage (C,(k + 1))) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
L~ (Cage (C,(k + 1))) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Upper_Arc (L~ (Cage (C,(k + 1)))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
(Upper_Appr C) . (k + 1) is functional Element of K636( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
South_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
x is set
x is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C ` is functional Element of K6( the carrier of (TOP-REAL 2))
the carrier of (Euclid 2) is non empty set
Lower_Appr C is V19() V22( NAT ) V23(K636( the carrier of (TOP-REAL 2))) Function-like V46( NAT ,K636( the carrier of (TOP-REAL 2))) Element of K6(K7(NAT,K636( the carrier of (TOP-REAL 2))))
Lim_inf (Lower_Appr C) is functional Element of K6( the carrier of (TOP-REAL 2))
BDD C is functional open Element of K6( the carrier of (TOP-REAL 2))
UBD C is non empty functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
(BDD C) \/ (UBD C) is non empty functional Element of K6( the carrier of (TOP-REAL 2))
the carrier of (TopSpaceMetr (Euclid 2)) is set
K6( the carrier of (TopSpaceMetr (Euclid 2))) is set
A is Element of K6( the carrier of (TopSpaceMetr (Euclid 2)))
e is Element of the carrier of (Euclid 2)
m is V11() real ext-real set
Ball (e,m) is Element of K6( the carrier of (Euclid 2))
K6( the carrier of (Euclid 2)) is set
OO is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
OO + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,(OO + 1)) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,(OO + 1))) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,(OO + 1)))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
(Lower_Appr C) . (OO + 1) is functional Element of K636( the carrier of (TOP-REAL 2))
OO + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
UBD-Family C is non empty Element of K6(K6( the carrier of (TOP-REAL 2)))
union (UBD-Family C) is set
A is set
{ (UBD (L~ (Cage (C,b1)))) where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT : verum } is set
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,m) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
L~ (Cage (C,m)) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
UBD (L~ (Cage (C,m))) is non empty functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
the carrier of (TopSpaceMetr (Euclid 2)) is set
K6( the carrier of (TopSpaceMetr (Euclid 2))) is set
LeftComp (Cage (C,m)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
OO is Element of K6( the carrier of (TopSpaceMetr (Euclid 2)))
e is Element of the carrier of (Euclid 2)
r is V11() real ext-real set
Ball (e,r) is Element of K6( the carrier of (Euclid 2))
K6( the carrier of (Euclid 2)) is set
k is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
(Lower_Appr C) . m is functional Element of K636( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,m))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
Cage (C,k) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
LeftComp (Cage (C,k)) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
k + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() Element of NAT
Cage (C,(k + 1)) is non empty V2() V19() V22( NAT ) V23( the carrier of (TOP-REAL 2)) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() FinSequence of the carrier of (TOP-REAL 2)
LeftComp (Cage (C,(k + 1))) is functional open connected V224( TOP-REAL 2) Element of K6( the carrier of (TOP-REAL 2))
L~ (Cage (C,(k + 1))) is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
Lower_Arc (L~ (Cage (C,(k + 1)))) is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
(Lower_Appr C) . (k + 1) is functional Element of K636( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
LMP C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
E-bound C is V11() real ext-real Element of REAL
W-bound C is V11() real ext-real Element of REAL
(E-bound C) + (W-bound C) is V11() real ext-real Element of REAL
((E-bound C) + (W-bound C)) / 2 is V11() real ext-real Element of REAL
Vertical_Line (((E-bound C) + (W-bound C)) / 2) is functional Element of K6( the carrier of (TOP-REAL 2))
{ b1 where b1 is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2) : b1 `1 = ((E-bound C) + (W-bound C)) / 2 } is set
C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)) is functional Element of K6( the carrier of (TOP-REAL 2))
proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2))) is V158() V159() V160() Element of K6(REAL)
lower_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound C) + (W-bound C)) / 2),(lower_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Lower_Arc C is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
UMP C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
upper_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))) is V11() real ext-real Element of REAL
|[(((E-bound C) + (W-bound C)) / 2),(upper_bound (proj2 .: (C /\ (Vertical_Line (((E-bound C) + (W-bound C)) / 2)))))]| is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
Upper_Arc C is non empty functional closed connected bounded compact with_the_max_arc Element of K6( the carrier of (TOP-REAL 2))
South_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
North_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
W-bound C is V11() real ext-real Element of REAL
North_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
W-bound (North_Arc C) is V11() real ext-real Element of REAL
W-min C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
E-bound C is V11() real ext-real Element of REAL
North_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
E-bound (North_Arc C) is V11() real ext-real Element of REAL
E-max C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
W-bound C is V11() real ext-real Element of REAL
South_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
W-bound (South_Arc C) is V11() real ext-real Element of REAL
W-min C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)
C is non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical Element of K6( the carrier of (TOP-REAL 2))
E-bound C is V11() real ext-real Element of REAL
South_Arc C is functional closed bounded compact Element of K6( the carrier of (TOP-REAL 2))
E-bound (South_Arc C) is V11() real ext-real Element of REAL
E-max C is V19() Function-like V41(2) FinSequence-like V148() V149() V150() Element of the carrier of (TOP-REAL 2)