REAL is non empty non trivial complex-membered ext-real-membered real-membered V56() non finite non bounded_below non bounded_above V71() set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() non finite left_end bounded_below cardinal limit_cardinal Element of bool REAL
bool REAL is non empty non trivial non finite set
COMPLEX is non empty non trivial complex-membered V56() non finite set
RAT is non empty non trivial complex-membered ext-real-membered real-membered rational-membered V56() non finite set
INT is non empty non trivial complex-membered ext-real-membered real-membered rational-membered integer-membered V56() non finite set
[:COMPLEX,COMPLEX:] is Relation-like non empty non trivial complex-valued non finite set
bool [:COMPLEX,COMPLEX:] is non empty non trivial non finite set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like non empty non trivial complex-valued non finite set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial non finite set
[:REAL,REAL:] is Relation-like non empty non trivial complex-valued ext-real-valued real-valued non finite set
bool [:REAL,REAL:] is non empty non trivial non finite set
[:[:REAL,REAL:],REAL:] is Relation-like non empty non trivial complex-valued ext-real-valued real-valued non finite set
bool [:[:REAL,REAL:],REAL:] is non empty non trivial non finite set
[:RAT,RAT:] is Relation-like RAT -valued non empty non trivial complex-valued ext-real-valued real-valued non finite set
bool [:RAT,RAT:] is non empty non trivial non finite set
[:[:RAT,RAT:],RAT:] is Relation-like RAT -valued non empty non trivial complex-valued ext-real-valued real-valued non finite set
bool [:[:RAT,RAT:],RAT:] is non empty non trivial non finite set
[:INT,INT:] is Relation-like RAT -valued INT -valued non empty non trivial complex-valued ext-real-valued real-valued non finite set
bool [:INT,INT:] is non empty non trivial non finite set
[:[:INT,INT:],INT:] is Relation-like RAT -valued INT -valued non empty non trivial complex-valued ext-real-valued real-valued non finite set
bool [:[:INT,INT:],INT:] is non empty non trivial non finite set
[:NAT,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial complex-valued ext-real-valued real-valued natural-valued non finite set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued non empty non trivial complex-valued ext-real-valued real-valued natural-valued non finite set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite set
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
the Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
omega is non empty non trivial epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() non finite left_end bounded_below cardinal limit_cardinal set
bool omega is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
ExtREAL is non empty ext-real-membered V71() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of NAT
2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
3 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
[:NAT,REAL:] is Relation-like non empty non trivial complex-valued ext-real-valued real-valued non finite set
bool [:NAT,REAL:] is non empty non trivial non finite set
bool [:NAT,NAT:] is non empty non trivial non finite set
dom {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
rng {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty trivial Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
|.0.| is complex real ext-real V49() Element of REAL
Re 0 is complex real ext-real Element of REAL
(Re 0) ^2 is complex real ext-real Element of REAL
(Re 0) * (Re 0) is complex real ext-real set
Im 0 is complex real ext-real Element of REAL
(Im 0) ^2 is complex real ext-real Element of REAL
(Im 0) * (Im 0) is complex real ext-real set
((Re 0) ^2) + ((Im 0) ^2) is complex real ext-real Element of REAL
sqrt (((Re 0) ^2) + ((Im 0) ^2)) is complex real ext-real Element of REAL
sqrt 0 is complex real ext-real Element of REAL
compcomplex is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
addcomplex is Relation-like [:COMPLEX,COMPLEX:] -defined COMPLEX -valued Function-like non empty total V18([:COMPLEX,COMPLEX:], COMPLEX ) complex-valued commutative associative V79( COMPLEX ) Element of bool [:[:COMPLEX,COMPLEX:],COMPLEX:]
diffcomplex is Relation-like [:COMPLEX,COMPLEX:] -defined COMPLEX -valued Function-like non empty total V18([:COMPLEX,COMPLEX:], COMPLEX ) complex-valued Element of bool [:[:COMPLEX,COMPLEX:],COMPLEX:]
multcomplex is Relation-like [:COMPLEX,COMPLEX:] -defined COMPLEX -valued Function-like non empty total V18([:COMPLEX,COMPLEX:], COMPLEX ) complex-valued commutative associative V79( COMPLEX ) Element of bool [:[:COMPLEX,COMPLEX:],COMPLEX:]
the_unity_wrt addcomplex is complex Element of COMPLEX
the_unity_wrt multcomplex is complex Element of COMPLEX
card {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
R is complex-membered ext-real-membered real-membered Element of bool REAL
r0 is complex-membered ext-real-membered real-membered Element of bool REAL
r1 is complex real ext-real set
t is complex real ext-real set
r1 is complex real ext-real set
t is complex real ext-real Element of REAL
t is complex real ext-real set
m is complex real ext-real set
R is complex real ext-real set
r0 is complex-membered ext-real-membered real-membered Element of bool REAL
r1 is complex-membered ext-real-membered real-membered Element of bool REAL
t is complex real ext-real set
t is complex real ext-real set
t + R is complex real ext-real Element of REAL
t + 0 is complex real ext-real Element of REAL
t is complex real ext-real set
t is complex real ext-real set
t is complex real ext-real set
t - R is complex real ext-real Element of REAL
m is complex real ext-real set
m + R is complex real ext-real Element of REAL
(t - R) + R is complex real ext-real Element of REAL
- 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of REAL
- R is complex real ext-real Element of REAL
t + 0 is complex real ext-real Element of REAL
t + (- R) is complex real ext-real Element of REAL
m is complex real ext-real set
R is complex real ext-real set
r0 is complex real ext-real set
r0 + 1 is complex real ext-real Element of REAL
r0 is complex real ext-real set
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R is complex-membered ext-real-membered real-membered Element of bool REAL
r0 is complex real ext-real set
r1 is complex real ext-real set
r1 is complex real ext-real set
t is complex real ext-real set
abs r0 is complex real ext-real Element of REAL
Re r0 is complex real ext-real Element of REAL
(Re r0) ^2 is complex real ext-real Element of REAL
(Re r0) * (Re r0) is complex real ext-real set
Im r0 is complex real ext-real Element of REAL
(Im r0) ^2 is complex real ext-real Element of REAL
(Im r0) * (Im r0) is complex real ext-real set
((Re r0) ^2) + ((Im r0) ^2) is complex real ext-real Element of REAL
sqrt (((Re r0) ^2) + ((Im r0) ^2)) is complex real ext-real Element of REAL
abs r1 is complex real ext-real Element of REAL
Re r1 is complex real ext-real Element of REAL
(Re r1) ^2 is complex real ext-real Element of REAL
(Re r1) * (Re r1) is complex real ext-real set
Im r1 is complex real ext-real Element of REAL
(Im r1) ^2 is complex real ext-real Element of REAL
(Im r1) * (Im r1) is complex real ext-real set
((Re r1) ^2) + ((Im r1) ^2) is complex real ext-real Element of REAL
sqrt (((Re r1) ^2) + ((Im r1) ^2)) is complex real ext-real Element of REAL
(abs r0) + (abs r1) is complex real ext-real Element of REAL
((abs r0) + (abs r1)) + 1 is complex real ext-real Element of REAL
t is complex real ext-real Element of REAL
t is complex real ext-real set
abs t is complex real ext-real Element of REAL
Re t is complex real ext-real Element of REAL
(Re t) ^2 is complex real ext-real Element of REAL
(Re t) * (Re t) is complex real ext-real set
Im t is complex real ext-real Element of REAL
(Im t) ^2 is complex real ext-real Element of REAL
(Im t) * (Im t) is complex real ext-real set
((Re t) ^2) + ((Im t) ^2) is complex real ext-real Element of REAL
sqrt (((Re t) ^2) + ((Im t) ^2)) is complex real ext-real Element of REAL
t + 0 is complex real ext-real Element of REAL
- (abs r1) is complex real ext-real Element of REAL
- (abs r0) is complex real ext-real Element of REAL
(- (abs r0)) + (- (abs r1)) is complex real ext-real Element of REAL
0 + t is complex real ext-real Element of REAL
(- (abs r0)) - (abs r1) is complex real ext-real Element of REAL
- 1 is complex real ext-real non positive V48() V49() Element of INT
((- (abs r0)) - (abs r1)) + (- 1) is complex real ext-real Element of REAL
((- (abs r0)) - (abs r1)) - 1 is complex real ext-real Element of REAL
- (((abs r0) + (abs r1)) + 1) is complex real ext-real Element of REAL
r0 is complex real ext-real set
- r0 is complex real ext-real Element of REAL
r1 is ext-real set
t is complex real ext-real set
abs t is complex real ext-real Element of REAL
Re t is complex real ext-real Element of REAL
(Re t) ^2 is complex real ext-real Element of REAL
(Re t) * (Re t) is complex real ext-real set
Im t is complex real ext-real Element of REAL
(Im t) ^2 is complex real ext-real Element of REAL
(Im t) * (Im t) is complex real ext-real set
((Re t) ^2) + ((Im t) ^2) is complex real ext-real Element of REAL
sqrt (((Re t) ^2) + ((Im t) ^2)) is complex real ext-real Element of REAL
- (abs t) is complex real ext-real Element of REAL
r1 is ext-real set
t is complex real ext-real set
abs t is complex real ext-real Element of REAL
Re t is complex real ext-real Element of REAL
(Re t) ^2 is complex real ext-real Element of REAL
(Re t) * (Re t) is complex real ext-real set
Im t is complex real ext-real Element of REAL
(Im t) ^2 is complex real ext-real Element of REAL
(Im t) * (Im t) is complex real ext-real set
((Re t) ^2) + ((Im t) ^2) is complex real ext-real Element of REAL
sqrt (((Re t) ^2) + ((Im t) ^2)) is complex real ext-real Element of REAL
R is complex real ext-real set
{R} is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element set
r0 is set
R is complex-membered ext-real-membered real-membered set
r0 is complex real ext-real set
r1 is complex real ext-real set
r1 is complex-membered ext-real-membered real-membered Element of bool REAL
t is complex real ext-real set
t is complex real ext-real set
t is complex real ext-real set
t is complex real ext-real Element of REAL
m is complex real ext-real Element of REAL
m is complex real ext-real set
m - m is complex real ext-real Element of REAL
f1 is complex real ext-real set
m - (m - m) is complex real ext-real Element of REAL
(m - m) - (m - m) is complex real ext-real Element of REAL
m is complex real ext-real set
f1 is complex real ext-real set
m - f1 is complex real ext-real Element of REAL
R is complex real ext-real set
r0 is complex real ext-real set
r1 is complex-membered ext-real-membered real-membered set
R - r0 is complex real ext-real Element of REAL
R - (R - r0) is complex real ext-real Element of REAL
t is complex real ext-real set
r0 - R is complex real ext-real Element of REAL
r0 - (r0 - R) is complex real ext-real Element of REAL
t is complex real ext-real set
R is complex-membered ext-real-membered real-membered set
r0 is complex real ext-real set
r1 is complex real ext-real set
r1 is complex-membered ext-real-membered real-membered Element of bool REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
t is complex real ext-real set
m is complex real ext-real set
t is complex real ext-real set
m is complex real ext-real Element of REAL
m is complex real ext-real Element of REAL
f1 is complex real ext-real set
m + f1 is complex real ext-real Element of REAL
f2 is complex real ext-real set
(m + f1) - m is complex real ext-real Element of REAL
m - m is complex real ext-real Element of REAL
f1 is complex real ext-real set
f2 is complex real ext-real set
m + f2 is complex real ext-real Element of REAL
R is complex real ext-real set
r0 is complex real ext-real set
r1 is complex-membered ext-real-membered real-membered set
R - r0 is complex real ext-real Element of REAL
r0 + (R - r0) is complex real ext-real Element of REAL
t is complex real ext-real set
r0 - R is complex real ext-real Element of REAL
R + (r0 - R) is complex real ext-real Element of REAL
t is complex real ext-real set
R is complex-membered ext-real-membered real-membered set
r0 is complex real ext-real set
r1 is complex real ext-real set
R is complex-membered ext-real-membered real-membered set
r0 is complex real ext-real set
r1 is complex real ext-real set
R is complex real ext-real set
r0 is non empty complex-membered ext-real-membered real-membered set
(r0) is complex real ext-real set
r1 is complex real ext-real set
R + r1 is complex real ext-real Element of REAL
r1 is ext-real set
R is non empty complex-membered ext-real-membered real-membered set
(R) is complex real ext-real set
r0 is complex real ext-real set
r1 is complex real ext-real set
r0 - r1 is complex real ext-real Element of REAL
r0 + r1 is complex real ext-real Element of REAL
r1 is ext-real set
R is non empty complex-membered ext-real-membered real-membered bounded_below set
(R) is complex real ext-real set
inf R is complex real ext-real set
r0 is complex real ext-real set
r1 is ext-real set
r0 is complex real ext-real set
R is non empty complex-membered ext-real-membered real-membered bounded_above set
(R) is complex real ext-real set
sup R is complex real ext-real set
r0 is complex real ext-real set
r1 is ext-real set
r0 is complex real ext-real set
R is complex-membered ext-real-membered real-membered Element of bool REAL
(R) is complex real ext-real set
(R) is complex real ext-real set
R is complex real ext-real set
(R) is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element Element of bool REAL
((R)) is complex real ext-real Element of REAL
inf (R) is complex real ext-real set
r0 is complex real ext-real set
(r0) is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element Element of bool REAL
((r0)) is complex real ext-real Element of REAL
sup (r0) is complex real ext-real set
R is complex real ext-real set
(R) is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element Element of bool REAL
((R)) is complex real ext-real Element of REAL
inf (R) is complex real ext-real set
((R)) is complex real ext-real Element of REAL
sup (R) is complex real ext-real set
R is complex-membered ext-real-membered real-membered Element of bool REAL
(R) is complex real ext-real Element of REAL
(R) is complex real ext-real Element of REAL
r0 is non empty complex-membered ext-real-membered real-membered bounded_below bounded_above real-bounded set
(r0) is complex real ext-real set
inf r0 is complex real ext-real set
(r0) is complex real ext-real set
sup r0 is complex real ext-real set
R is complex-membered ext-real-membered real-membered Element of bool REAL
(R) is complex real ext-real Element of REAL
(R) is complex real ext-real Element of REAL
r0 is complex real ext-real set
r1 is complex real ext-real set
r0 is complex real ext-real Element of REAL
r1 is set
t is set
(r0) is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element Element of bool REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
abs R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 is complex real ext-real set
r1 is complex real ext-real Element of REAL
abs r1 is complex real ext-real Element of REAL
Re r1 is complex real ext-real Element of REAL
(Re r1) ^2 is complex real ext-real Element of REAL
(Re r1) * (Re r1) is complex real ext-real set
Im r1 is complex real ext-real Element of REAL
(Im r1) ^2 is complex real ext-real Element of REAL
(Im r1) * (Im r1) is complex real ext-real set
((Re r1) ^2) + ((Im r1) ^2) is complex real ext-real Element of REAL
sqrt (((Re r1) ^2) + ((Im r1) ^2)) is complex real ext-real Element of REAL
t is complex real ext-real Element of REAL
t is complex real ext-real set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(abs R) . f1 is complex real ext-real Element of REAL
((abs R) . f1) - t is complex real ext-real Element of REAL
abs (((abs R) . f1) - t) is complex real ext-real Element of REAL
Re (((abs R) . f1) - t) is complex real ext-real Element of REAL
(Re (((abs R) . f1) - t)) ^2 is complex real ext-real Element of REAL
(Re (((abs R) . f1) - t)) * (Re (((abs R) . f1) - t)) is complex real ext-real set
Im (((abs R) . f1) - t) is complex real ext-real Element of REAL
(Im (((abs R) . f1) - t)) ^2 is complex real ext-real Element of REAL
(Im (((abs R) . f1) - t)) * (Im (((abs R) . f1) - t)) is complex real ext-real set
((Re (((abs R) . f1) - t)) ^2) + ((Im (((abs R) . f1) - t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((abs R) . f1) - t)) ^2) + ((Im (((abs R) . f1) - t)) ^2)) is complex real ext-real Element of REAL
R . f1 is complex real ext-real Element of REAL
r1 - (R . f1) is complex real ext-real Element of REAL
abs (r1 - (R . f1)) is complex real ext-real Element of REAL
Re (r1 - (R . f1)) is complex real ext-real Element of REAL
(Re (r1 - (R . f1))) ^2 is complex real ext-real Element of REAL
(Re (r1 - (R . f1))) * (Re (r1 - (R . f1))) is complex real ext-real set
Im (r1 - (R . f1)) is complex real ext-real Element of REAL
(Im (r1 - (R . f1))) ^2 is complex real ext-real Element of REAL
(Im (r1 - (R . f1))) * (Im (r1 - (R . f1))) is complex real ext-real set
((Re (r1 - (R . f1))) ^2) + ((Im (r1 - (R . f1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (r1 - (R . f1))) ^2) + ((Im (r1 - (R . f1))) ^2)) is complex real ext-real Element of REAL
(R . f1) - r1 is complex real ext-real Element of REAL
- ((R . f1) - r1) is complex real ext-real Element of REAL
abs (- ((R . f1) - r1)) is complex real ext-real Element of REAL
Re (- ((R . f1) - r1)) is complex real ext-real Element of REAL
(Re (- ((R . f1) - r1))) ^2 is complex real ext-real Element of REAL
(Re (- ((R . f1) - r1))) * (Re (- ((R . f1) - r1))) is complex real ext-real set
Im (- ((R . f1) - r1)) is complex real ext-real Element of REAL
(Im (- ((R . f1) - r1))) ^2 is complex real ext-real Element of REAL
(Im (- ((R . f1) - r1))) * (Im (- ((R . f1) - r1))) is complex real ext-real set
((Re (- ((R . f1) - r1))) ^2) + ((Im (- ((R . f1) - r1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (- ((R . f1) - r1))) ^2) + ((Im (- ((R . f1) - r1))) ^2)) is complex real ext-real Element of REAL
abs ((R . f1) - r1) is complex real ext-real Element of REAL
Re ((R . f1) - r1) is complex real ext-real Element of REAL
(Re ((R . f1) - r1)) ^2 is complex real ext-real Element of REAL
(Re ((R . f1) - r1)) * (Re ((R . f1) - r1)) is complex real ext-real set
Im ((R . f1) - r1) is complex real ext-real Element of REAL
(Im ((R . f1) - r1)) ^2 is complex real ext-real Element of REAL
(Im ((R . f1) - r1)) * (Im ((R . f1) - r1)) is complex real ext-real set
((Re ((R . f1) - r1)) ^2) + ((Im ((R . f1) - r1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . f1) - r1)) ^2) + ((Im ((R . f1) - r1)) ^2)) is complex real ext-real Element of REAL
abs (R . f1) is complex real ext-real Element of REAL
Re (R . f1) is complex real ext-real Element of REAL
(Re (R . f1)) ^2 is complex real ext-real Element of REAL
(Re (R . f1)) * (Re (R . f1)) is complex real ext-real set
Im (R . f1) is complex real ext-real Element of REAL
(Im (R . f1)) ^2 is complex real ext-real Element of REAL
(Im (R . f1)) * (Im (R . f1)) is complex real ext-real set
((Re (R . f1)) ^2) + ((Im (R . f1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (R . f1)) ^2) + ((Im (R . f1)) ^2)) is complex real ext-real Element of REAL
t - (abs (R . f1)) is complex real ext-real Element of REAL
(abs (R . f1)) - t is complex real ext-real Element of REAL
- (abs ((R . f1) - r1)) is complex real ext-real Element of REAL
- (t - (abs (R . f1))) is complex real ext-real Element of REAL
abs ((abs (R . f1)) - t) is complex real ext-real Element of REAL
Re ((abs (R . f1)) - t) is complex real ext-real Element of REAL
(Re ((abs (R . f1)) - t)) ^2 is complex real ext-real Element of REAL
(Re ((abs (R . f1)) - t)) * (Re ((abs (R . f1)) - t)) is complex real ext-real set
Im ((abs (R . f1)) - t) is complex real ext-real Element of REAL
(Im ((abs (R . f1)) - t)) ^2 is complex real ext-real Element of REAL
(Im ((abs (R . f1)) - t)) * (Im ((abs (R . f1)) - t)) is complex real ext-real set
((Re ((abs (R . f1)) - t)) ^2) + ((Im ((abs (R . f1)) - t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((abs (R . f1)) - t)) ^2) + ((Im ((abs (R . f1)) - t)) ^2)) is complex real ext-real Element of REAL
(abs R) . f1 is complex real ext-real Element of REAL
((abs R) . f1) - t is complex real ext-real Element of REAL
abs (((abs R) . f1) - t) is complex real ext-real Element of REAL
Re (((abs R) . f1) - t) is complex real ext-real Element of REAL
(Re (((abs R) . f1) - t)) ^2 is complex real ext-real Element of REAL
(Re (((abs R) . f1) - t)) * (Re (((abs R) . f1) - t)) is complex real ext-real set
Im (((abs R) . f1) - t) is complex real ext-real Element of REAL
(Im (((abs R) . f1) - t)) ^2 is complex real ext-real Element of REAL
(Im (((abs R) . f1) - t)) * (Im (((abs R) . f1) - t)) is complex real ext-real set
((Re (((abs R) . f1) - t)) ^2) + ((Im (((abs R) . f1) - t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((abs R) . f1) - t)) ^2) + ((Im (((abs R) . f1) - t)) ^2)) is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued bounded_above bounded_below bounded convergent Element of bool [:NAT,REAL:]
abs R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
abs R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim (abs R) is complex real ext-real Element of REAL
lim R is complex real ext-real Element of REAL
abs (lim R) is complex real ext-real Element of REAL
Re (lim R) is complex real ext-real Element of REAL
(Re (lim R)) ^2 is complex real ext-real Element of REAL
(Re (lim R)) * (Re (lim R)) is complex real ext-real set
Im (lim R) is complex real ext-real Element of REAL
(Im (lim R)) ^2 is complex real ext-real Element of REAL
(Im (lim R)) * (Im (lim R)) is complex real ext-real set
((Re (lim R)) ^2) + ((Im (lim R)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (lim R)) ^2) + ((Im (lim R)) ^2)) is complex real ext-real Element of REAL
r0 is complex real ext-real set
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . t is complex real ext-real Element of REAL
(lim R) - (R . t) is complex real ext-real Element of REAL
abs ((lim R) - (R . t)) is complex real ext-real Element of REAL
Re ((lim R) - (R . t)) is complex real ext-real Element of REAL
(Re ((lim R) - (R . t))) ^2 is complex real ext-real Element of REAL
(Re ((lim R) - (R . t))) * (Re ((lim R) - (R . t))) is complex real ext-real set
Im ((lim R) - (R . t)) is complex real ext-real Element of REAL
(Im ((lim R) - (R . t))) ^2 is complex real ext-real Element of REAL
(Im ((lim R) - (R . t))) * (Im ((lim R) - (R . t))) is complex real ext-real set
((Re ((lim R) - (R . t))) ^2) + ((Im ((lim R) - (R . t))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((lim R) - (R . t))) ^2) + ((Im ((lim R) - (R . t))) ^2)) is complex real ext-real Element of REAL
(R . t) - (lim R) is complex real ext-real Element of REAL
- ((R . t) - (lim R)) is complex real ext-real Element of REAL
abs (- ((R . t) - (lim R))) is complex real ext-real Element of REAL
Re (- ((R . t) - (lim R))) is complex real ext-real Element of REAL
(Re (- ((R . t) - (lim R)))) ^2 is complex real ext-real Element of REAL
(Re (- ((R . t) - (lim R)))) * (Re (- ((R . t) - (lim R)))) is complex real ext-real set
Im (- ((R . t) - (lim R))) is complex real ext-real Element of REAL
(Im (- ((R . t) - (lim R)))) ^2 is complex real ext-real Element of REAL
(Im (- ((R . t) - (lim R)))) * (Im (- ((R . t) - (lim R)))) is complex real ext-real set
((Re (- ((R . t) - (lim R)))) ^2) + ((Im (- ((R . t) - (lim R)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (- ((R . t) - (lim R)))) ^2) + ((Im (- ((R . t) - (lim R)))) ^2)) is complex real ext-real Element of REAL
abs ((R . t) - (lim R)) is complex real ext-real Element of REAL
Re ((R . t) - (lim R)) is complex real ext-real Element of REAL
(Re ((R . t) - (lim R))) ^2 is complex real ext-real Element of REAL
(Re ((R . t) - (lim R))) * (Re ((R . t) - (lim R))) is complex real ext-real set
Im ((R . t) - (lim R)) is complex real ext-real Element of REAL
(Im ((R . t) - (lim R))) ^2 is complex real ext-real Element of REAL
(Im ((R . t) - (lim R))) * (Im ((R . t) - (lim R))) is complex real ext-real set
((Re ((R . t) - (lim R))) ^2) + ((Im ((R . t) - (lim R))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . t) - (lim R))) ^2) + ((Im ((R . t) - (lim R))) ^2)) is complex real ext-real Element of REAL
abs (R . t) is complex real ext-real Element of REAL
Re (R . t) is complex real ext-real Element of REAL
(Re (R . t)) ^2 is complex real ext-real Element of REAL
(Re (R . t)) * (Re (R . t)) is complex real ext-real set
Im (R . t) is complex real ext-real Element of REAL
(Im (R . t)) ^2 is complex real ext-real Element of REAL
(Im (R . t)) * (Im (R . t)) is complex real ext-real set
((Re (R . t)) ^2) + ((Im (R . t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (R . t)) ^2) + ((Im (R . t)) ^2)) is complex real ext-real Element of REAL
(abs (lim R)) - (abs (R . t)) is complex real ext-real Element of REAL
(abs (R . t)) - (abs (lim R)) is complex real ext-real Element of REAL
- (abs ((R . t) - (lim R))) is complex real ext-real Element of REAL
- ((abs (lim R)) - (abs (R . t))) is complex real ext-real Element of REAL
abs ((abs (R . t)) - (abs (lim R))) is complex real ext-real Element of REAL
Re ((abs (R . t)) - (abs (lim R))) is complex real ext-real Element of REAL
(Re ((abs (R . t)) - (abs (lim R)))) ^2 is complex real ext-real Element of REAL
(Re ((abs (R . t)) - (abs (lim R)))) * (Re ((abs (R . t)) - (abs (lim R)))) is complex real ext-real set
Im ((abs (R . t)) - (abs (lim R))) is complex real ext-real Element of REAL
(Im ((abs (R . t)) - (abs (lim R)))) ^2 is complex real ext-real Element of REAL
(Im ((abs (R . t)) - (abs (lim R)))) * (Im ((abs (R . t)) - (abs (lim R)))) is complex real ext-real set
((Re ((abs (R . t)) - (abs (lim R)))) ^2) + ((Im ((abs (R . t)) - (abs (lim R)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((abs (R . t)) - (abs (lim R)))) ^2) + ((Im ((abs (R . t)) - (abs (lim R)))) ^2)) is complex real ext-real Element of REAL
(abs R) . t is complex real ext-real Element of REAL
((abs R) . t) - (abs (lim R)) is complex real ext-real Element of REAL
abs (((abs R) . t) - (abs (lim R))) is complex real ext-real Element of REAL
Re (((abs R) . t) - (abs (lim R))) is complex real ext-real Element of REAL
(Re (((abs R) . t) - (abs (lim R)))) ^2 is complex real ext-real Element of REAL
(Re (((abs R) . t) - (abs (lim R)))) * (Re (((abs R) . t) - (abs (lim R)))) is complex real ext-real set
Im (((abs R) . t) - (abs (lim R))) is complex real ext-real Element of REAL
(Im (((abs R) . t) - (abs (lim R)))) ^2 is complex real ext-real Element of REAL
(Im (((abs R) . t) - (abs (lim R)))) * (Im (((abs R) . t) - (abs (lim R)))) is complex real ext-real set
((Re (((abs R) . t) - (abs (lim R)))) ^2) + ((Im (((abs R) . t) - (abs (lim R)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((abs R) . t) - (abs (lim R)))) ^2) + ((Im (((abs R) . t) - (abs (lim R)))) ^2)) is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
abs R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim (abs R) is complex real ext-real Element of REAL
lim R is complex real ext-real Element of REAL
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . r0 is complex real ext-real Element of REAL
abs (R . r0) is complex real ext-real Element of REAL
Re (R . r0) is complex real ext-real Element of REAL
(Re (R . r0)) ^2 is complex real ext-real Element of REAL
(Re (R . r0)) * (Re (R . r0)) is complex real ext-real set
Im (R . r0) is complex real ext-real Element of REAL
(Im (R . r0)) ^2 is complex real ext-real Element of REAL
(Im (R . r0)) * (Im (R . r0)) is complex real ext-real set
((Re (R . r0)) ^2) + ((Im (R . r0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (R . r0)) ^2) + ((Im (R . r0)) ^2)) is complex real ext-real Element of REAL
- (abs (R . r0)) is complex real ext-real Element of REAL
(abs R) . r0 is complex real ext-real Element of REAL
- ((abs R) . r0) is complex real ext-real Element of REAL
- (abs R) is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
- 1 is complex real ext-real non positive set
(- 1) (#) (abs R) is Relation-like NAT -defined Function-like non empty total complex-valued ext-real-valued real-valued set
(- (abs R)) . r0 is complex real ext-real Element of REAL
lim (- (abs R)) is complex real ext-real Element of REAL
- (lim (abs R)) is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r1 is complex real ext-real set
t is complex real ext-real set
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
(R . m) - r1 is complex real ext-real Element of REAL
abs ((R . m) - r1) is complex real ext-real Element of REAL
Re ((R . m) - r1) is complex real ext-real Element of REAL
(Re ((R . m) - r1)) ^2 is complex real ext-real Element of REAL
(Re ((R . m) - r1)) * (Re ((R . m) - r1)) is complex real ext-real set
Im ((R . m) - r1) is complex real ext-real Element of REAL
(Im ((R . m) - r1)) ^2 is complex real ext-real Element of REAL
(Im ((R . m) - r1)) * (Im ((R . m) - r1)) is complex real ext-real set
((Re ((R . m) - r1)) ^2) + ((Im ((R . m) - r1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . m) - r1)) ^2) + ((Im ((R . m) - r1)) ^2)) is complex real ext-real Element of REAL
f1 is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
r0 * f1 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of r0
f1 . m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
r0 . (f1 . m) is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim R is complex real ext-real Element of REAL
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim r0 is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
r0 * r1 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of r0
t is complex real ext-real set
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r1 . m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
r0 . (r1 . m) is complex real ext-real Element of REAL
(R . m) - (lim r0) is complex real ext-real Element of REAL
abs ((R . m) - (lim r0)) is complex real ext-real Element of REAL
Re ((R . m) - (lim r0)) is complex real ext-real Element of REAL
(Re ((R . m) - (lim r0))) ^2 is complex real ext-real Element of REAL
(Re ((R . m) - (lim r0))) * (Re ((R . m) - (lim r0))) is complex real ext-real set
Im ((R . m) - (lim r0)) is complex real ext-real Element of REAL
(Im ((R . m) - (lim r0))) ^2 is complex real ext-real Element of REAL
(Im ((R . m) - (lim r0))) * (Im ((R . m) - (lim r0))) is complex real ext-real set
((Re ((R . m) - (lim r0))) ^2) + ((Im ((R . m) - (lim r0))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . m) - (lim r0))) ^2) + ((Im ((R . m) - (lim r0))) ^2)) is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r1 is complex real ext-real set
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is complex real ext-real set
m is complex real ext-real set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . f2 is complex real ext-real Element of REAL
(R . f2) - r1 is complex real ext-real Element of REAL
abs ((R . f2) - r1) is complex real ext-real Element of REAL
Re ((R . f2) - r1) is complex real ext-real Element of REAL
(Re ((R . f2) - r1)) ^2 is complex real ext-real Element of REAL
(Re ((R . f2) - r1)) * (Re ((R . f2) - r1)) is complex real ext-real set
Im ((R . f2) - r1) is complex real ext-real Element of REAL
(Im ((R . f2) - r1)) ^2 is complex real ext-real Element of REAL
(Im ((R . f2) - r1)) * (Im ((R . f2) - r1)) is complex real ext-real set
((Re ((R . f2) - r1)) ^2) + ((Im ((R . f2) - r1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . f2) - r1)) ^2) + ((Im ((R . f2) - r1)) ^2)) is complex real ext-real Element of REAL
r0 . f2 is complex real ext-real Element of REAL
(r0 . f2) - t is complex real ext-real Element of REAL
abs ((r0 . f2) - t) is complex real ext-real Element of REAL
Re ((r0 . f2) - t) is complex real ext-real Element of REAL
(Re ((r0 . f2) - t)) ^2 is complex real ext-real Element of REAL
(Re ((r0 . f2) - t)) * (Re ((r0 . f2) - t)) is complex real ext-real set
Im ((r0 . f2) - t) is complex real ext-real Element of REAL
(Im ((r0 . f2) - t)) ^2 is complex real ext-real Element of REAL
(Im ((r0 . f2) - t)) * (Im ((r0 . f2) - t)) is complex real ext-real set
((Re ((r0 . f2) - t)) ^2) + ((Im ((r0 . f2) - t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . f2) - t)) ^2) + ((Im ((r0 . f2) - t)) ^2)) is complex real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . f2 is complex real ext-real Element of REAL
R . f2 is complex real ext-real Element of REAL
(r0 . f2) - t is complex real ext-real Element of REAL
abs ((r0 . f2) - t) is complex real ext-real Element of REAL
Re ((r0 . f2) - t) is complex real ext-real Element of REAL
(Re ((r0 . f2) - t)) ^2 is complex real ext-real Element of REAL
(Re ((r0 . f2) - t)) * (Re ((r0 . f2) - t)) is complex real ext-real set
Im ((r0 . f2) - t) is complex real ext-real Element of REAL
(Im ((r0 . f2) - t)) ^2 is complex real ext-real Element of REAL
(Im ((r0 . f2) - t)) * (Im ((r0 . f2) - t)) is complex real ext-real set
((Re ((r0 . f2) - t)) ^2) + ((Im ((r0 . f2) - t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . f2) - t)) ^2) + ((Im ((r0 . f2) - t)) ^2)) is complex real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim R is complex real ext-real Element of REAL
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim r0 is complex real ext-real Element of REAL
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is complex real ext-real set
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
(R . m) - (lim R) is complex real ext-real Element of REAL
abs ((R . m) - (lim R)) is complex real ext-real Element of REAL
Re ((R . m) - (lim R)) is complex real ext-real Element of REAL
(Re ((R . m) - (lim R))) ^2 is complex real ext-real Element of REAL
(Re ((R . m) - (lim R))) * (Re ((R . m) - (lim R))) is complex real ext-real set
Im ((R . m) - (lim R)) is complex real ext-real Element of REAL
(Im ((R . m) - (lim R))) ^2 is complex real ext-real Element of REAL
(Im ((R . m) - (lim R))) * (Im ((R . m) - (lim R))) is complex real ext-real set
((Re ((R . m) - (lim R))) ^2) + ((Im ((R . m) - (lim R))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . m) - (lim R))) ^2) + ((Im ((R . m) - (lim R))) ^2)) is complex real ext-real Element of REAL
r0 . m is complex real ext-real Element of REAL
(r0 . m) - (lim R) is complex real ext-real Element of REAL
abs ((r0 . m) - (lim R)) is complex real ext-real Element of REAL
Re ((r0 . m) - (lim R)) is complex real ext-real Element of REAL
(Re ((r0 . m) - (lim R))) ^2 is complex real ext-real Element of REAL
(Re ((r0 . m) - (lim R))) * (Re ((r0 . m) - (lim R))) is complex real ext-real set
Im ((r0 . m) - (lim R)) is complex real ext-real Element of REAL
(Im ((r0 . m) - (lim R))) ^2 is complex real ext-real Element of REAL
(Im ((r0 . m) - (lim R))) * (Im ((r0 . m) - (lim R))) is complex real ext-real set
((Re ((r0 . m) - (lim R))) ^2) + ((Im ((r0 . m) - (lim R))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . m) - (lim R))) ^2) + ((Im ((r0 . m) - (lim R))) ^2)) is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . m is complex real ext-real Element of REAL
R . m is complex real ext-real Element of REAL
(r0 . m) - (lim R) is complex real ext-real Element of REAL
abs ((r0 . m) - (lim R)) is complex real ext-real Element of REAL
Re ((r0 . m) - (lim R)) is complex real ext-real Element of REAL
(Re ((r0 . m) - (lim R))) ^2 is complex real ext-real Element of REAL
(Re ((r0 . m) - (lim R))) * (Re ((r0 . m) - (lim R))) is complex real ext-real set
Im ((r0 . m) - (lim R)) is complex real ext-real Element of REAL
(Im ((r0 . m) - (lim R))) ^2 is complex real ext-real Element of REAL
(Im ((r0 . m) - (lim R))) * (Im ((r0 . m) - (lim R))) is complex real ext-real set
((Re ((r0 . m) - (lim R))) ^2) + ((Im ((r0 . m) - (lim R))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . m) - (lim R))) ^2) + ((Im ((r0 . m) - (lim R))) ^2)) is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
the Relation-like NAT -defined REAL -valued Function-like constant non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued non-decreasing non-increasing bounded_above bounded_below bounded convergent Element of bool [:NAT,REAL:] is Relation-like NAT -defined REAL -valued Function-like constant non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued non-decreasing non-increasing bounded_above bounded_below bounded convergent Element of bool [:NAT,REAL:]
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued bounded_above bounded_below bounded convergent Element of bool [:NAT,REAL:]
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R ^\ r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
r1 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 ^\ R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of r0
lim (r0 ^\ R) is complex real ext-real Element of REAL
lim r0 is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 ^\ R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of r0
r1 is complex real ext-real set
t is complex real ext-real set
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . m is complex real ext-real Element of REAL
(r0 . m) - r1 is complex real ext-real Element of REAL
abs ((r0 . m) - r1) is complex real ext-real Element of REAL
Re ((r0 . m) - r1) is complex real ext-real Element of REAL
(Re ((r0 . m) - r1)) ^2 is complex real ext-real Element of REAL
(Re ((r0 . m) - r1)) * (Re ((r0 . m) - r1)) is complex real ext-real set
Im ((r0 . m) - r1) is complex real ext-real Element of REAL
(Im ((r0 . m) - r1)) ^2 is complex real ext-real Element of REAL
(Im ((r0 . m) - r1)) * (Im ((r0 . m) - r1)) is complex real ext-real set
((Re ((r0 . m) - r1)) ^2) + ((Im ((r0 . m) - r1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . m) - r1)) ^2) + ((Im ((r0 . m) - r1)) ^2)) is complex real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
(t + R) + f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m - R is complex real ext-real V48() V49() Element of INT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(t + f2) + R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
- R is complex real ext-real non positive V48() V49() Element of INT
((t + f2) + R) + (- R) is complex real ext-real V48() V49() Element of INT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f1 + R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0 ^\ R) . f1 is complex real ext-real Element of REAL
((r0 ^\ R) . f1) - r1 is complex real ext-real Element of REAL
abs (((r0 ^\ R) . f1) - r1) is complex real ext-real Element of REAL
Re (((r0 ^\ R) . f1) - r1) is complex real ext-real Element of REAL
(Re (((r0 ^\ R) . f1) - r1)) ^2 is complex real ext-real Element of REAL
(Re (((r0 ^\ R) . f1) - r1)) * (Re (((r0 ^\ R) . f1) - r1)) is complex real ext-real set
Im (((r0 ^\ R) . f1) - r1) is complex real ext-real Element of REAL
(Im (((r0 ^\ R) . f1) - r1)) ^2 is complex real ext-real Element of REAL
(Im (((r0 ^\ R) . f1) - r1)) * (Im (((r0 ^\ R) . f1) - r1)) is complex real ext-real set
((Re (((r0 ^\ R) . f1) - r1)) ^2) + ((Im (((r0 ^\ R) . f1) - r1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((r0 ^\ R) . f1) - r1)) ^2) + ((Im (((r0 ^\ R) . f1) - r1)) ^2)) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 ^\ R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of r0
lim r0 is complex real ext-real Element of REAL
lim (r0 ^\ R) is complex real ext-real Element of REAL
r1 is complex real ext-real set
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
(t + R) + m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m - R is complex real ext-real V48() V49() Element of INT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(t + f1) + R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
- R is complex real ext-real non positive V48() V49() Element of INT
((t + f1) + R) + (- R) is complex real ext-real V48() V49() Element of INT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 + R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0 ^\ R) . f2 is complex real ext-real Element of REAL
((r0 ^\ R) . f2) - (lim (r0 ^\ R)) is complex real ext-real Element of REAL
abs (((r0 ^\ R) . f2) - (lim (r0 ^\ R))) is complex real ext-real Element of REAL
Re (((r0 ^\ R) . f2) - (lim (r0 ^\ R))) is complex real ext-real Element of REAL
(Re (((r0 ^\ R) . f2) - (lim (r0 ^\ R)))) ^2 is complex real ext-real Element of REAL
(Re (((r0 ^\ R) . f2) - (lim (r0 ^\ R)))) * (Re (((r0 ^\ R) . f2) - (lim (r0 ^\ R)))) is complex real ext-real set
Im (((r0 ^\ R) . f2) - (lim (r0 ^\ R))) is complex real ext-real Element of REAL
(Im (((r0 ^\ R) . f2) - (lim (r0 ^\ R)))) ^2 is complex real ext-real Element of REAL
(Im (((r0 ^\ R) . f2) - (lim (r0 ^\ R)))) * (Im (((r0 ^\ R) . f2) - (lim (r0 ^\ R)))) is complex real ext-real set
((Re (((r0 ^\ R) . f2) - (lim (r0 ^\ R)))) ^2) + ((Im (((r0 ^\ R) . f2) - (lim (r0 ^\ R)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((r0 ^\ R) . f2) - (lim (r0 ^\ R)))) ^2) + ((Im (((r0 ^\ R) . f2) - (lim (r0 ^\ R)))) ^2)) is complex real ext-real Element of REAL
r0 . m is complex real ext-real Element of REAL
(r0 . m) - (lim (r0 ^\ R)) is complex real ext-real Element of REAL
abs ((r0 . m) - (lim (r0 ^\ R))) is complex real ext-real Element of REAL
Re ((r0 . m) - (lim (r0 ^\ R))) is complex real ext-real Element of REAL
(Re ((r0 . m) - (lim (r0 ^\ R)))) ^2 is complex real ext-real Element of REAL
(Re ((r0 . m) - (lim (r0 ^\ R)))) * (Re ((r0 . m) - (lim (r0 ^\ R)))) is complex real ext-real set
Im ((r0 . m) - (lim (r0 ^\ R))) is complex real ext-real Element of REAL
(Im ((r0 . m) - (lim (r0 ^\ R)))) ^2 is complex real ext-real Element of REAL
(Im ((r0 . m) - (lim (r0 ^\ R)))) * (Im ((r0 . m) - (lim (r0 ^\ R)))) is complex real ext-real set
((Re ((r0 . m) - (lim (r0 ^\ R)))) ^2) + ((Im ((r0 . m) - (lim (r0 ^\ R)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . m) - (lim (r0 ^\ R)))) ^2) + ((Im ((r0 . m) - (lim (r0 ^\ R)))) ^2)) is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim R is complex real ext-real Element of REAL
abs (lim R) is complex real ext-real Element of REAL
Re (lim R) is complex real ext-real Element of REAL
(Re (lim R)) ^2 is complex real ext-real Element of REAL
(Re (lim R)) * (Re (lim R)) is complex real ext-real set
Im (lim R) is complex real ext-real Element of REAL
(Im (lim R)) ^2 is complex real ext-real Element of REAL
(Im (lim R)) * (Im (lim R)) is complex real ext-real set
((Re (lim R)) ^2) + ((Im (lim R)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (lim R)) ^2) + ((Im (lim R)) ^2)) is complex real ext-real Element of REAL
(abs (lim R)) / 2 is complex real ext-real Element of REAL
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R ^\ r1 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
0 + r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . (t + r1) is complex real ext-real Element of REAL
abs (R . (t + r1)) is complex real ext-real Element of REAL
Re (R . (t + r1)) is complex real ext-real Element of REAL
(Re (R . (t + r1))) ^2 is complex real ext-real Element of REAL
(Re (R . (t + r1))) * (Re (R . (t + r1))) is complex real ext-real set
Im (R . (t + r1)) is complex real ext-real Element of REAL
(Im (R . (t + r1))) ^2 is complex real ext-real Element of REAL
(Im (R . (t + r1))) * (Im (R . (t + r1))) is complex real ext-real set
((Re (R . (t + r1))) ^2) + ((Im (R . (t + r1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (R . (t + r1))) ^2) + ((Im (R . (t + r1))) ^2)) is complex real ext-real Element of REAL
(R ^\ r1) . t is complex real ext-real Element of REAL
abs ((R ^\ r1) . t) is complex real ext-real Element of REAL
Re ((R ^\ r1) . t) is complex real ext-real Element of REAL
(Re ((R ^\ r1) . t)) ^2 is complex real ext-real Element of REAL
(Re ((R ^\ r1) . t)) * (Re ((R ^\ r1) . t)) is complex real ext-real set
Im ((R ^\ r1) . t) is complex real ext-real Element of REAL
(Im ((R ^\ r1) . t)) ^2 is complex real ext-real Element of REAL
(Im ((R ^\ r1) . t)) * (Im ((R ^\ r1) . t)) is complex real ext-real set
((Re ((R ^\ r1) . t)) ^2) + ((Im ((R ^\ r1) . t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R ^\ r1) . t)) ^2) + ((Im ((R ^\ r1) . t)) ^2)) is complex real ext-real Element of REAL
0 / 2 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of RAT
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim R is complex real ext-real Element of REAL
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R ^\ r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
R is complex real ext-real set
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
rng r0 is non empty complex-membered ext-real-membered real-membered Element of bool REAL
lim r0 is complex real ext-real Element of REAL
r1 is complex real ext-real Element of REAL
(r1) is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element Element of bool REAL
t is complex real ext-real Element of REAL
t is complex real ext-real set
m is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom r0 is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of bool NAT
r0 . m is complex real ext-real Element of REAL
(r0 . m) - R is complex real ext-real Element of REAL
abs ((r0 . m) - R) is complex real ext-real Element of REAL
Re ((r0 . m) - R) is complex real ext-real Element of REAL
(Re ((r0 . m) - R)) ^2 is complex real ext-real Element of REAL
(Re ((r0 . m) - R)) * (Re ((r0 . m) - R)) is complex real ext-real set
Im ((r0 . m) - R) is complex real ext-real Element of REAL
(Im ((r0 . m) - R)) ^2 is complex real ext-real Element of REAL
(Im ((r0 . m) - R)) * (Im ((r0 . m) - R)) is complex real ext-real set
((Re ((r0 . m) - R)) ^2) + ((Im ((r0 . m) - R)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . m) - R)) ^2) + ((Im ((r0 . m) - R)) ^2)) is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . t is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . t is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . t is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . t is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . t is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim R is complex real ext-real Element of REAL
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . r0 is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim R is complex real ext-real Element of REAL
(lim R) " is complex real ext-real Element of REAL
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 " is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim (r0 ") is complex real ext-real Element of REAL
lim r0 is complex real ext-real Element of REAL
R is complex real ext-real set
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r1 is complex real ext-real set
r1 " is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + R is complex real ext-real Element of REAL
(r1 ") + 0 is complex real ext-real Element of REAL
1 / (r1 ") is complex real ext-real Element of REAL
1 / (t + R) is complex real ext-real Element of REAL
(r1 ") " is complex real ext-real Element of REAL
1 * ((r1 ") ") is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . m is complex real ext-real Element of REAL
(r0 . m) - 0 is complex real ext-real Element of REAL
abs ((r0 . m) - 0) is complex real ext-real Element of REAL
Re ((r0 . m) - 0) is complex real ext-real Element of REAL
(Re ((r0 . m) - 0)) ^2 is complex real ext-real Element of REAL
(Re ((r0 . m) - 0)) * (Re ((r0 . m) - 0)) is complex real ext-real set
Im ((r0 . m) - 0) is complex real ext-real Element of REAL
(Im ((r0 . m) - 0)) ^2 is complex real ext-real Element of REAL
(Im ((r0 . m) - 0)) * (Im ((r0 . m) - 0)) is complex real ext-real set
((Re ((r0 . m) - 0)) ^2) + ((Im ((r0 . m) - 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . m) - 0)) ^2) + ((Im ((r0 . m) - 0)) ^2)) is complex real ext-real Element of REAL
t + R is complex real ext-real Element of REAL
m + R is complex real ext-real Element of REAL
1 / (m + R) is complex real ext-real Element of REAL
1 / (t + R) is complex real ext-real Element of REAL
r0 . m is complex real ext-real Element of REAL
R is complex real ext-real set
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim r0 is complex real ext-real Element of REAL
r1 is complex real ext-real set
r1 " is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + R is complex real ext-real Element of REAL
(r1 ") + 0 is complex real ext-real Element of REAL
1 / (r1 ") is complex real ext-real Element of REAL
1 / (t + R) is complex real ext-real Element of REAL
(r1 ") " is complex real ext-real Element of REAL
1 * ((r1 ") ") is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + R is complex real ext-real Element of REAL
m + R is complex real ext-real Element of REAL
1 / (m + R) is complex real ext-real Element of REAL
1 / (t + R) is complex real ext-real Element of REAL
r0 . m is complex real ext-real Element of REAL
(r0 . m) - 0 is complex real ext-real Element of REAL
abs ((r0 . m) - 0) is complex real ext-real Element of REAL
Re ((r0 . m) - 0) is complex real ext-real Element of REAL
(Re ((r0 . m) - 0)) ^2 is complex real ext-real Element of REAL
(Re ((r0 . m) - 0)) * (Re ((r0 . m) - 0)) is complex real ext-real set
Im ((r0 . m) - 0) is complex real ext-real Element of REAL
(Im ((r0 . m) - 0)) ^2 is complex real ext-real Element of REAL
(Im ((r0 . m) - 0)) * (Im ((r0 . m) - 0)) is complex real ext-real set
((Re ((r0 . m) - 0)) ^2) + ((Im ((r0 . m) - 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . m) - 0)) ^2) + ((Im ((r0 . m) - 0)) ^2)) is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim R is complex real ext-real Element of REAL
R is complex real ext-real set
r0 is complex real ext-real set
r1 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim r1 is complex real ext-real Element of REAL
t is complex real ext-real Element of REAL
t is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 (#) t is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0 (#) t) . m is complex real ext-real Element of REAL
t . m is complex real ext-real Element of REAL
r0 * (t . m) is complex real ext-real Element of REAL
m + R is complex real ext-real Element of REAL
1 / (m + R) is complex real ext-real Element of REAL
r0 * (1 / (m + R)) is complex real ext-real Element of REAL
(m + R) " is complex real ext-real Element of REAL
1 * ((m + R) ") is complex real ext-real Element of REAL
r0 * (1 * ((m + R) ")) is complex real ext-real Element of REAL
r0 / (m + R) is complex real ext-real Element of REAL
r1 . m is complex real ext-real Element of REAL
lim (r0 (#) t) is complex real ext-real Element of REAL
lim t is complex real ext-real Element of REAL
r0 * (lim t) is complex real ext-real Element of REAL
r0 * 0 is complex real ext-real Element of REAL
R is complex real ext-real set
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r1 is complex real ext-real set
r1 " is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
t * t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + R is complex real ext-real Element of REAL
(t * t) + R is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . m is complex real ext-real Element of REAL
(r0 . m) - 0 is complex real ext-real Element of REAL
abs ((r0 . m) - 0) is complex real ext-real Element of REAL
Re ((r0 . m) - 0) is complex real ext-real Element of REAL
(Re ((r0 . m) - 0)) ^2 is complex real ext-real Element of REAL
(Re ((r0 . m) - 0)) * (Re ((r0 . m) - 0)) is complex real ext-real set
Im ((r0 . m) - 0) is complex real ext-real Element of REAL
(Im ((r0 . m) - 0)) ^2 is complex real ext-real Element of REAL
(Im ((r0 . m) - 0)) * (Im ((r0 . m) - 0)) is complex real ext-real set
((Re ((r0 . m) - 0)) ^2) + ((Im ((r0 . m) - 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . m) - 0)) ^2) + ((Im ((r0 . m) - 0)) ^2)) is complex real ext-real Element of REAL
t * t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m * m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(t * t) + R is complex real ext-real Element of REAL
(m * m) + R is complex real ext-real Element of REAL
(r1 ") + 0 is complex real ext-real Element of REAL
1 / (r1 ") is complex real ext-real Element of REAL
1 / ((t * t) + R) is complex real ext-real Element of REAL
(r1 ") " is complex real ext-real Element of REAL
1 * ((r1 ") ") is complex real ext-real Element of REAL
t ^2 is complex real ext-real Element of REAL
t * t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
1 / ((m * m) + R) is complex real ext-real Element of REAL
1 / ((t * t) + R) is complex real ext-real Element of REAL
r0 . m is complex real ext-real Element of REAL
R is complex real ext-real set
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim r0 is complex real ext-real Element of REAL
r1 is complex real ext-real set
r1 " is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
t * t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + R is complex real ext-real Element of REAL
(t * t) + R is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t * t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m * m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(t * t) + R is complex real ext-real Element of REAL
(m * m) + R is complex real ext-real Element of REAL
(r1 ") + 0 is complex real ext-real Element of REAL
1 / (r1 ") is complex real ext-real Element of REAL
1 / ((t * t) + R) is complex real ext-real Element of REAL
(r1 ") " is complex real ext-real Element of REAL
1 * ((r1 ") ") is complex real ext-real Element of REAL
t ^2 is complex real ext-real Element of REAL
t * t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
1 / ((m * m) + R) is complex real ext-real Element of REAL
1 / ((t * t) + R) is complex real ext-real Element of REAL
r0 . m is complex real ext-real Element of REAL
(r0 . m) - 0 is complex real ext-real Element of REAL
abs ((r0 . m) - 0) is complex real ext-real Element of REAL
Re ((r0 . m) - 0) is complex real ext-real Element of REAL
(Re ((r0 . m) - 0)) ^2 is complex real ext-real Element of REAL
(Re ((r0 . m) - 0)) * (Re ((r0 . m) - 0)) is complex real ext-real set
Im ((r0 . m) - 0) is complex real ext-real Element of REAL
(Im ((r0 . m) - 0)) ^2 is complex real ext-real Element of REAL
(Im ((r0 . m) - 0)) * (Im ((r0 . m) - 0)) is complex real ext-real set
((Re ((r0 . m) - 0)) ^2) + ((Im ((r0 . m) - 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((r0 . m) - 0)) ^2) + ((Im ((r0 . m) - 0)) ^2)) is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim R is complex real ext-real Element of REAL
R is complex real ext-real set
r0 is complex real ext-real set
r1 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim r1 is complex real ext-real Element of REAL
t is complex real ext-real Element of REAL
t is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 (#) t is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0 (#) t) . m is complex real ext-real Element of REAL
t . m is complex real ext-real Element of REAL
r0 * (t . m) is complex real ext-real Element of REAL
m * m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(m * m) + R is complex real ext-real Element of REAL
1 / ((m * m) + R) is complex real ext-real Element of REAL
r0 * (1 / ((m * m) + R)) is complex real ext-real Element of REAL
((m * m) + R) " is complex real ext-real Element of REAL
1 * (((m * m) + R) ") is complex real ext-real Element of REAL
r0 * (1 * (((m * m) + R) ")) is complex real ext-real Element of REAL
r0 / ((m * m) + R) is complex real ext-real Element of REAL
r1 . m is complex real ext-real Element of REAL
lim (r0 (#) t) is complex real ext-real Element of REAL
lim t is complex real ext-real Element of REAL
r0 * (lim t) is complex real ext-real Element of REAL
r0 * 0 is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 is complex real ext-real set
r1 is complex-membered ext-real-membered real-membered Element of bool REAL
t is complex real ext-real set
t is complex real ext-real set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
t is complex real ext-real set
t is ext-real set
(r1) is complex real ext-real Element of REAL
t is complex real ext-real Element of REAL
t is complex real ext-real set
R . 0 is complex real ext-real Element of REAL
(r1) - t is complex real ext-real Element of REAL
m is complex real ext-real set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . f2 is complex real ext-real Element of REAL
(R . f2) - t is complex real ext-real Element of REAL
abs ((R . f2) - t) is complex real ext-real Element of REAL
Re ((R . f2) - t) is complex real ext-real Element of REAL
(Re ((R . f2) - t)) ^2 is complex real ext-real Element of REAL
(Re ((R . f2) - t)) * (Re ((R . f2) - t)) is complex real ext-real set
Im ((R . f2) - t) is complex real ext-real Element of REAL
(Im ((R . f2) - t)) ^2 is complex real ext-real Element of REAL
(Im ((R . f2) - t)) * (Im ((R . f2) - t)) is complex real ext-real set
((Re ((R . f2) - t)) ^2) + ((Im ((R . f2) - t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . f2) - t)) ^2) + ((Im ((R . f2) - t)) ^2)) is complex real ext-real Element of REAL
R . f1 is complex real ext-real Element of REAL
R . f2 is complex real ext-real Element of REAL
- t is complex real ext-real Element of REAL
t + (- t) is complex real ext-real Element of REAL
(R . f2) - t is complex real ext-real Element of REAL
f1 is complex real ext-real set
t + t is complex real ext-real Element of REAL
(R . f2) + 0 is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 is complex-membered ext-real-membered real-membered Element of bool REAL
(r0) is complex real ext-real Element of REAL
r1 is complex real ext-real Element of REAL
t is complex real ext-real set
t is complex real ext-real set
m is ext-real set
R . 0 is complex real ext-real Element of REAL
(r0) + t is complex real ext-real Element of REAL
t is complex real ext-real set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . f1 is complex real ext-real Element of REAL
(R . f1) - r1 is complex real ext-real Element of REAL
abs ((R . f1) - r1) is complex real ext-real Element of REAL
Re ((R . f1) - r1) is complex real ext-real Element of REAL
(Re ((R . f1) - r1)) ^2 is complex real ext-real Element of REAL
(Re ((R . f1) - r1)) * (Re ((R . f1) - r1)) is complex real ext-real set
Im ((R . f1) - r1) is complex real ext-real Element of REAL
(Im ((R . f1) - r1)) ^2 is complex real ext-real Element of REAL
(Im ((R . f1) - r1)) * (Im ((R . f1) - r1)) is complex real ext-real set
((Re ((R . f1) - r1)) ^2) + ((Im ((R . f1) - r1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . f1) - r1)) ^2) + ((Im ((R . f1) - r1)) ^2)) is complex real ext-real Element of REAL
f2 is complex real ext-real set
f2 is complex real ext-real set
f1 is complex real ext-real set
k is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . k is complex real ext-real Element of REAL
R . f1 is complex real ext-real Element of REAL
0 + r1 is complex real ext-real Element of REAL
f1 is complex real ext-real set
(R . f1) - r1 is complex real ext-real Element of REAL
R . m is complex real ext-real Element of REAL
r1 + t is complex real ext-real Element of REAL
- 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of REAL
- t is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim R is complex real ext-real Element of REAL
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . r0 is complex real ext-real Element of REAL
NAT --> (R . r0) is Relation-like NAT -defined REAL -valued Function-like constant non empty total V18( NAT , REAL ) T-Sequence-like complex-valued ext-real-valued real-valued non-decreasing non-increasing bounded_above bounded_below bounded convergent Element of bool [:NAT,REAL:]
t is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(NAT --> (R . r0)) . t is complex real ext-real Element of REAL
t . t is complex real ext-real Element of REAL
r0 + t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . (r0 + t) is complex real ext-real Element of REAL
(NAT --> (R . r0)) . 0 is complex real ext-real Element of REAL
lim (NAT --> (R . r0)) is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
t . t is complex real ext-real Element of REAL
t + r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . (t + r0) is complex real ext-real Element of REAL
R ^\ r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
lim t is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim R is complex real ext-real Element of REAL
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . r0 is complex real ext-real Element of REAL
NAT --> (R . r0) is Relation-like NAT -defined REAL -valued Function-like constant non empty total V18( NAT , REAL ) T-Sequence-like complex-valued ext-real-valued real-valued non-decreasing non-increasing bounded_above bounded_below bounded convergent Element of bool [:NAT,REAL:]
t is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(NAT --> (R . r0)) . t is complex real ext-real Element of REAL
t . t is complex real ext-real Element of REAL
r0 + t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . (r0 + t) is complex real ext-real Element of REAL
(NAT --> (R . r0)) . 0 is complex real ext-real Element of REAL
lim (NAT --> (R . r0)) is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
t . t is complex real ext-real Element of REAL
t + r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . (t + r0) is complex real ext-real Element of REAL
R ^\ r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
lim t is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
1 + r1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
R ^\ (1 + r1) is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . t is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
(t + 1) + r1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
0 + r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . ((t + 1) + r1) is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
((t + 1) + r1) + m is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
m - (1 + r1) is complex real ext-real V48() V49() Element of INT
t + 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(t + 0) + f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + (1 + r1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
R . (t + (1 + r1)) is complex real ext-real Element of REAL
(R ^\ (1 + r1)) . t is complex real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(m - (1 + r1)) + (1 + r1) is complex real ext-real V48() V49() Element of INT
f1 + (1 + r1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
(R ^\ (1 + r1)) . f1 is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R ^\ (1 + r1)) . m is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R ^\ (1 + r1)) . m is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
(R ^\ (1 + r1)) . m is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
(R ^\ (1 + r1)) . m is complex real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R ^\ (1 + r1)) . f2 is complex real ext-real Element of REAL
(R ^\ (1 + r1)) . f1 is complex real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R ^\ (1 + r1)) . f1 is complex real ext-real Element of REAL
t is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued Element of bool [:NAT,NAT:]
t . 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom t is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of bool NAT
rng t is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of bool REAL
m is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m . m is complex real ext-real Element of REAL
rng m is non empty complex-membered ext-real-membered real-membered Element of bool REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m . m is complex real ext-real Element of REAL
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
m . (m + 1) is complex real ext-real Element of REAL
f1 is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
R * f1 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
m is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
f1 * m is Relation-like NAT -defined NAT -valued RAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing subsequence of f1
f2 is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
m . (f2 + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R ^\ (1 + r1)) . (m . (f2 + 1)) is complex real ext-real Element of REAL
m . f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R ^\ (1 + r1)) . (m . f2) is complex real ext-real Element of REAL
(R ^\ (1 + r1)) * m is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R ^\ (1 + r1)
((R ^\ (1 + r1)) * m) . (f2 + 1) is complex real ext-real Element of REAL
(R * f1) * m is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R * f1
((R * f1) * m) . (f2 + 1) is complex real ext-real Element of REAL
((R ^\ (1 + r1)) * m) . f2 is complex real ext-real Element of REAL
f1 is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
R * f1 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
(R * f1) . (f2 + 1) is complex real ext-real Element of REAL
((R * f1) * m) . f2 is complex real ext-real Element of REAL
(R * f1) . f2 is complex real ext-real Element of REAL
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued Element of bool [:NAT,NAT:]
t . 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom t is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of bool NAT
rng t is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of bool REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
m . m is complex real ext-real Element of REAL
rng m is non empty complex-membered ext-real-membered real-membered Element of bool REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m . m is complex real ext-real Element of REAL
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
m . (m + 1) is complex real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
m . f1 is complex real ext-real Element of REAL
f1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
m . (f1 + 1) is complex real ext-real Element of REAL
m . 0 is complex real ext-real Element of REAL
m is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
f1 is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f1 . f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
f1 . (f2 + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . (f1 . (f2 + 1)) is complex real ext-real Element of REAL
R . (f1 . f2) is complex real ext-real Element of REAL
R * f1 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
(R * f1) . f2 is complex real ext-real Element of REAL
(R * f1) . (f2 + 1) is complex real ext-real Element of REAL
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
R * t is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
t is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
R * t is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
R * r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
r1 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
r0 is complex real ext-real set
r1 is complex real ext-real set
r1 / 2 is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . t is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . m is complex real ext-real Element of REAL
(R . m) - (R . t) is complex real ext-real Element of REAL
abs ((R . m) - (R . t)) is complex real ext-real Element of REAL
Re ((R . m) - (R . t)) is complex real ext-real Element of REAL
(Re ((R . m) - (R . t))) ^2 is complex real ext-real Element of REAL
(Re ((R . m) - (R . t))) * (Re ((R . m) - (R . t))) is complex real ext-real set
Im ((R . m) - (R . t)) is complex real ext-real Element of REAL
(Im ((R . m) - (R . t))) ^2 is complex real ext-real Element of REAL
(Im ((R . m) - (R . t))) * (Im ((R . m) - (R . t))) is complex real ext-real set
((Re ((R . m) - (R . t))) ^2) + ((Im ((R . m) - (R . t))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . m) - (R . t))) ^2) + ((Im ((R . m) - (R . t))) ^2)) is complex real ext-real Element of REAL
(R . m) - r0 is complex real ext-real Element of REAL
abs ((R . m) - r0) is complex real ext-real Element of REAL
Re ((R . m) - r0) is complex real ext-real Element of REAL
(Re ((R . m) - r0)) ^2 is complex real ext-real Element of REAL
(Re ((R . m) - r0)) * (Re ((R . m) - r0)) is complex real ext-real set
Im ((R . m) - r0) is complex real ext-real Element of REAL
(Im ((R . m) - r0)) ^2 is complex real ext-real Element of REAL
(Im ((R . m) - r0)) * (Im ((R . m) - r0)) is complex real ext-real set
((Re ((R . m) - r0)) ^2) + ((Im ((R . m) - r0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . m) - r0)) ^2) + ((Im ((R . m) - r0)) ^2)) is complex real ext-real Element of REAL
r0 - (R . t) is complex real ext-real Element of REAL
((R . m) - r0) + (r0 - (R . t)) is complex real ext-real Element of REAL
abs (((R . m) - r0) + (r0 - (R . t))) is complex real ext-real Element of REAL
Re (((R . m) - r0) + (r0 - (R . t))) is complex real ext-real Element of REAL
(Re (((R . m) - r0) + (r0 - (R . t)))) ^2 is complex real ext-real Element of REAL
(Re (((R . m) - r0) + (r0 - (R . t)))) * (Re (((R . m) - r0) + (r0 - (R . t)))) is complex real ext-real set
Im (((R . m) - r0) + (r0 - (R . t))) is complex real ext-real Element of REAL
(Im (((R . m) - r0) + (r0 - (R . t)))) ^2 is complex real ext-real Element of REAL
(Im (((R . m) - r0) + (r0 - (R . t)))) * (Im (((R . m) - r0) + (r0 - (R . t)))) is complex real ext-real set
((Re (((R . m) - r0) + (r0 - (R . t)))) ^2) + ((Im (((R . m) - r0) + (r0 - (R . t)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((R . m) - r0) + (r0 - (R . t)))) ^2) + ((Im (((R . m) - r0) + (r0 - (R . t)))) ^2)) is complex real ext-real Element of REAL
abs (r0 - (R . t)) is complex real ext-real Element of REAL
Re (r0 - (R . t)) is complex real ext-real Element of REAL
(Re (r0 - (R . t))) ^2 is complex real ext-real Element of REAL
(Re (r0 - (R . t))) * (Re (r0 - (R . t))) is complex real ext-real set
Im (r0 - (R . t)) is complex real ext-real Element of REAL
(Im (r0 - (R . t))) ^2 is complex real ext-real Element of REAL
(Im (r0 - (R . t))) * (Im (r0 - (R . t))) is complex real ext-real set
((Re (r0 - (R . t))) ^2) + ((Im (r0 - (R . t))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (r0 - (R . t))) ^2) + ((Im (r0 - (R . t))) ^2)) is complex real ext-real Element of REAL
(abs ((R . m) - r0)) + (abs (r0 - (R . t))) is complex real ext-real Element of REAL
(R . t) - r0 is complex real ext-real Element of REAL
abs ((R . t) - r0) is complex real ext-real Element of REAL
Re ((R . t) - r0) is complex real ext-real Element of REAL
(Re ((R . t) - r0)) ^2 is complex real ext-real Element of REAL
(Re ((R . t) - r0)) * (Re ((R . t) - r0)) is complex real ext-real set
Im ((R . t) - r0) is complex real ext-real Element of REAL
(Im ((R . t) - r0)) ^2 is complex real ext-real Element of REAL
(Im ((R . t) - r0)) * (Im ((R . t) - r0)) is complex real ext-real set
((Re ((R . t) - r0)) ^2) + ((Im ((R . t) - r0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . t) - r0)) ^2) + ((Im ((R . t) - r0)) ^2)) is complex real ext-real Element of REAL
- (r0 - (R . t)) is complex real ext-real Element of REAL
abs (- (r0 - (R . t))) is complex real ext-real Element of REAL
Re (- (r0 - (R . t))) is complex real ext-real Element of REAL
(Re (- (r0 - (R . t)))) ^2 is complex real ext-real Element of REAL
(Re (- (r0 - (R . t)))) * (Re (- (r0 - (R . t)))) is complex real ext-real set
Im (- (r0 - (R . t))) is complex real ext-real Element of REAL
(Im (- (r0 - (R . t)))) ^2 is complex real ext-real Element of REAL
(Im (- (r0 - (R . t)))) * (Im (- (r0 - (R . t)))) is complex real ext-real set
((Re (- (r0 - (R . t)))) ^2) + ((Im (- (r0 - (R . t)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (- (r0 - (R . t)))) ^2) + ((Im (- (r0 - (R . t)))) ^2)) is complex real ext-real Element of REAL
(r1 / 2) + (r1 / 2) is complex real ext-real Element of REAL
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . r0 is complex real ext-real Element of REAL
r1 is complex real ext-real set
abs (R . r0) is complex real ext-real Element of REAL
Re (R . r0) is complex real ext-real Element of REAL
(Re (R . r0)) ^2 is complex real ext-real Element of REAL
(Re (R . r0)) * (Re (R . r0)) is complex real ext-real set
Im (R . r0) is complex real ext-real Element of REAL
(Im (R . r0)) ^2 is complex real ext-real Element of REAL
(Im (R . r0)) * (Im (R . r0)) is complex real ext-real set
((Re (R . r0)) ^2) + ((Im (R . r0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (R . r0)) ^2) + ((Im (R . r0)) ^2)) is complex real ext-real Element of REAL
r1 + (abs (R . r0)) is complex real ext-real Element of REAL
(r1 + (abs (R . r0))) + 1 is complex real ext-real Element of REAL
0 + 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of NAT
t is complex real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . t is complex real ext-real Element of REAL
(R . t) - (R . r0) is complex real ext-real Element of REAL
abs ((R . t) - (R . r0)) is complex real ext-real Element of REAL
Re ((R . t) - (R . r0)) is complex real ext-real Element of REAL
(Re ((R . t) - (R . r0))) ^2 is complex real ext-real Element of REAL
(Re ((R . t) - (R . r0))) * (Re ((R . t) - (R . r0))) is complex real ext-real set
Im ((R . t) - (R . r0)) is complex real ext-real Element of REAL
(Im ((R . t) - (R . r0))) ^2 is complex real ext-real Element of REAL
(Im ((R . t) - (R . r0))) * (Im ((R . t) - (R . r0))) is complex real ext-real set
((Re ((R . t) - (R . r0))) ^2) + ((Im ((R . t) - (R . r0))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . t) - (R . r0))) ^2) + ((Im ((R . t) - (R . r0))) ^2)) is complex real ext-real Element of REAL
abs (R . t) is complex real ext-real Element of REAL
Re (R . t) is complex real ext-real Element of REAL
(Re (R . t)) ^2 is complex real ext-real Element of REAL
(Re (R . t)) * (Re (R . t)) is complex real ext-real set
Im (R . t) is complex real ext-real Element of REAL
(Im (R . t)) ^2 is complex real ext-real Element of REAL
(Im (R . t)) * (Im (R . t)) is complex real ext-real set
((Re (R . t)) ^2) + ((Im (R . t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (R . t)) ^2) + ((Im (R . t)) ^2)) is complex real ext-real Element of REAL
(abs (R . t)) - (abs (R . r0)) is complex real ext-real Element of REAL
1 + (abs (R . r0)) is complex real ext-real Element of REAL
((abs (R . t)) - (abs (R . r0))) + (abs (R . r0)) is complex real ext-real Element of REAL
(abs (R . r0)) + 1 is complex real ext-real Element of REAL
r1 + ((abs (R . r0)) + 1) is complex real ext-real Element of REAL
0 + (abs (R . t)) is complex real ext-real Element of REAL
(abs (R . t)) + 0 is complex real ext-real Element of REAL
t is complex real ext-real Element of REAL
t is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
t is complex real ext-real set
m is complex real ext-real set
m / 3 is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . f1 is complex real ext-real Element of REAL
m + f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . f1 is complex real ext-real Element of REAL
(R . f1) - t is complex real ext-real Element of REAL
abs ((R . f1) - t) is complex real ext-real Element of REAL
Re ((R . f1) - t) is complex real ext-real Element of REAL
(Re ((R . f1) - t)) ^2 is complex real ext-real Element of REAL
(Re ((R . f1) - t)) * (Re ((R . f1) - t)) is complex real ext-real set
Im ((R . f1) - t) is complex real ext-real Element of REAL
(Im ((R . f1) - t)) ^2 is complex real ext-real Element of REAL
(Im ((R . f1) - t)) * (Im ((R . f1) - t)) is complex real ext-real set
((Re ((R . f1) - t)) ^2) + ((Im ((R . f1) - t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . f1) - t)) ^2) + ((Im ((R . f1) - t)) ^2)) is complex real ext-real Element of REAL
f2 is Relation-like NAT -defined NAT -valued Function-like non empty total V18( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of bool [:NAT,NAT:]
R * f2 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued subsequence of R
(R * f2) . f1 is complex real ext-real Element of REAL
((R * f2) . f1) - t is complex real ext-real Element of REAL
abs (((R * f2) . f1) - t) is complex real ext-real Element of REAL
Re (((R * f2) . f1) - t) is complex real ext-real Element of REAL
(Re (((R * f2) . f1) - t)) ^2 is complex real ext-real Element of REAL
(Re (((R * f2) . f1) - t)) * (Re (((R * f2) . f1) - t)) is complex real ext-real set
Im (((R * f2) . f1) - t) is complex real ext-real Element of REAL
(Im (((R * f2) . f1) - t)) ^2 is complex real ext-real Element of REAL
(Im (((R * f2) . f1) - t)) * (Im (((R * f2) . f1) - t)) is complex real ext-real set
((Re (((R * f2) . f1) - t)) ^2) + ((Im (((R * f2) . f1) - t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((R * f2) . f1) - t)) ^2) + ((Im (((R * f2) . f1) - t)) ^2)) is complex real ext-real Element of REAL
f2 . f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . (f2 . f1) is complex real ext-real Element of REAL
(R . (f2 . f1)) - t is complex real ext-real Element of REAL
abs ((R . (f2 . f1)) - t) is complex real ext-real Element of REAL
Re ((R . (f2 . f1)) - t) is complex real ext-real Element of REAL
(Re ((R . (f2 . f1)) - t)) ^2 is complex real ext-real Element of REAL
(Re ((R . (f2 . f1)) - t)) * (Re ((R . (f2 . f1)) - t)) is complex real ext-real set
Im ((R . (f2 . f1)) - t) is complex real ext-real Element of REAL
(Im ((R . (f2 . f1)) - t)) ^2 is complex real ext-real Element of REAL
(Im ((R . (f2 . f1)) - t)) * (Im ((R . (f2 . f1)) - t)) is complex real ext-real set
((Re ((R . (f2 . f1)) - t)) ^2) + ((Im ((R . (f2 . f1)) - t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . (f2 . f1)) - t)) ^2) + ((Im ((R . (f2 . f1)) - t)) ^2)) is complex real ext-real Element of REAL
(R . (f2 . f1)) - (R . f1) is complex real ext-real Element of REAL
abs ((R . (f2 . f1)) - (R . f1)) is complex real ext-real Element of REAL
Re ((R . (f2 . f1)) - (R . f1)) is complex real ext-real Element of REAL
(Re ((R . (f2 . f1)) - (R . f1))) ^2 is complex real ext-real Element of REAL
(Re ((R . (f2 . f1)) - (R . f1))) * (Re ((R . (f2 . f1)) - (R . f1))) is complex real ext-real set
Im ((R . (f2 . f1)) - (R . f1)) is complex real ext-real Element of REAL
(Im ((R . (f2 . f1)) - (R . f1))) ^2 is complex real ext-real Element of REAL
(Im ((R . (f2 . f1)) - (R . f1))) * (Im ((R . (f2 . f1)) - (R . f1))) is complex real ext-real set
((Re ((R . (f2 . f1)) - (R . f1))) ^2) + ((Im ((R . (f2 . f1)) - (R . f1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . (f2 . f1)) - (R . f1))) ^2) + ((Im ((R . (f2 . f1)) - (R . f1))) ^2)) is complex real ext-real Element of REAL
(R . f1) - (R . (f2 . f1)) is complex real ext-real Element of REAL
- ((R . f1) - (R . (f2 . f1))) is complex real ext-real Element of REAL
abs (- ((R . f1) - (R . (f2 . f1)))) is complex real ext-real Element of REAL
Re (- ((R . f1) - (R . (f2 . f1)))) is complex real ext-real Element of REAL
(Re (- ((R . f1) - (R . (f2 . f1))))) ^2 is complex real ext-real Element of REAL
(Re (- ((R . f1) - (R . (f2 . f1))))) * (Re (- ((R . f1) - (R . (f2 . f1))))) is complex real ext-real set
Im (- ((R . f1) - (R . (f2 . f1)))) is complex real ext-real Element of REAL
(Im (- ((R . f1) - (R . (f2 . f1))))) ^2 is complex real ext-real Element of REAL
(Im (- ((R . f1) - (R . (f2 . f1))))) * (Im (- ((R . f1) - (R . (f2 . f1))))) is complex real ext-real set
((Re (- ((R . f1) - (R . (f2 . f1))))) ^2) + ((Im (- ((R . f1) - (R . (f2 . f1))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (- ((R . f1) - (R . (f2 . f1))))) ^2) + ((Im (- ((R . f1) - (R . (f2 . f1))))) ^2)) is complex real ext-real Element of REAL
abs ((R . f1) - (R . (f2 . f1))) is complex real ext-real Element of REAL
Re ((R . f1) - (R . (f2 . f1))) is complex real ext-real Element of REAL
(Re ((R . f1) - (R . (f2 . f1)))) ^2 is complex real ext-real Element of REAL
(Re ((R . f1) - (R . (f2 . f1)))) * (Re ((R . f1) - (R . (f2 . f1)))) is complex real ext-real set
Im ((R . f1) - (R . (f2 . f1))) is complex real ext-real Element of REAL
(Im ((R . f1) - (R . (f2 . f1)))) ^2 is complex real ext-real Element of REAL
(Im ((R . f1) - (R . (f2 . f1)))) * (Im ((R . f1) - (R . (f2 . f1)))) is complex real ext-real set
((Re ((R . f1) - (R . (f2 . f1)))) ^2) + ((Im ((R . f1) - (R . (f2 . f1)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . f1) - (R . (f2 . f1)))) ^2) + ((Im ((R . f1) - (R . (f2 . f1)))) ^2)) is complex real ext-real Element of REAL
R . f1 is complex real ext-real Element of REAL
(R . f1) - (R . f1) is complex real ext-real Element of REAL
abs ((R . f1) - (R . f1)) is complex real ext-real Element of REAL
Re ((R . f1) - (R . f1)) is complex real ext-real Element of REAL
(Re ((R . f1) - (R . f1))) ^2 is complex real ext-real Element of REAL
(Re ((R . f1) - (R . f1))) * (Re ((R . f1) - (R . f1))) is complex real ext-real set
Im ((R . f1) - (R . f1)) is complex real ext-real Element of REAL
(Im ((R . f1) - (R . f1))) ^2 is complex real ext-real Element of REAL
(Im ((R . f1) - (R . f1))) * (Im ((R . f1) - (R . f1))) is complex real ext-real set
((Re ((R . f1) - (R . f1))) ^2) + ((Im ((R . f1) - (R . f1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . f1) - (R . f1))) ^2) + ((Im ((R . f1) - (R . f1))) ^2)) is complex real ext-real Element of REAL
(m / 3) + (m / 3) is complex real ext-real Element of REAL
(abs ((R . f1) - (R . f1))) + (abs ((R . f1) - (R . (f2 . f1)))) is complex real ext-real Element of REAL
((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))) is complex real ext-real Element of REAL
abs (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1)))) is complex real ext-real Element of REAL
Re (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1)))) is complex real ext-real Element of REAL
(Re (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))))) ^2 is complex real ext-real Element of REAL
(Re (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))))) * (Re (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))))) is complex real ext-real set
Im (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1)))) is complex real ext-real Element of REAL
(Im (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))))) ^2 is complex real ext-real Element of REAL
(Im (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))))) * (Im (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))))) is complex real ext-real set
((Re (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))))) ^2) + ((Im (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))))) ^2) + ((Im (((R . f1) - (R . f1)) + ((R . f1) - (R . (f2 . f1))))) ^2)) is complex real ext-real Element of REAL
(R . f1) - (R . (f2 . f1)) is complex real ext-real Element of REAL
abs ((R . f1) - (R . (f2 . f1))) is complex real ext-real Element of REAL
Re ((R . f1) - (R . (f2 . f1))) is complex real ext-real Element of REAL
(Re ((R . f1) - (R . (f2 . f1)))) ^2 is complex real ext-real Element of REAL
(Re ((R . f1) - (R . (f2 . f1)))) * (Re ((R . f1) - (R . (f2 . f1)))) is complex real ext-real set
Im ((R . f1) - (R . (f2 . f1))) is complex real ext-real Element of REAL
(Im ((R . f1) - (R . (f2 . f1)))) ^2 is complex real ext-real Element of REAL
(Im ((R . f1) - (R . (f2 . f1)))) * (Im ((R . f1) - (R . (f2 . f1)))) is complex real ext-real set
((Re ((R . f1) - (R . (f2 . f1)))) ^2) + ((Im ((R . f1) - (R . (f2 . f1)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((R . f1) - (R . (f2 . f1)))) ^2) + ((Im ((R . f1) - (R . (f2 . f1)))) ^2)) is complex real ext-real Element of REAL
((m / 3) + (m / 3)) + (m / 3) is complex real ext-real Element of REAL
(abs ((R . f1) - (R . (f2 . f1)))) + (abs ((R . (f2 . f1)) - t)) is complex real ext-real Element of REAL
((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t) is complex real ext-real Element of REAL
abs (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t)) is complex real ext-real Element of REAL
Re (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t)) is complex real ext-real Element of REAL
(Re (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t))) ^2 is complex real ext-real Element of REAL
(Re (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t))) * (Re (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t))) is complex real ext-real set
Im (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t)) is complex real ext-real Element of REAL
(Im (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t))) ^2 is complex real ext-real Element of REAL
(Im (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t))) * (Im (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t))) is complex real ext-real set
((Re (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t))) ^2) + ((Im (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t))) ^2) + ((Im (((R . f1) - (R . (f2 . f1))) + ((R . (f2 . f1)) - t))) ^2)) is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
R . 0 is complex real ext-real Element of REAL
r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
R + r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim (R + r0) is complex real ext-real Element of REAL
lim r0 is complex real ext-real Element of REAL
(R . 0) + (lim r0) is complex real ext-real Element of REAL
R - r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
- r0 is Relation-like NAT -defined Function-like non empty total complex-valued ext-real-valued real-valued set
- 1 is complex real ext-real non positive set
(- 1) (#) r0 is Relation-like NAT -defined Function-like non empty total complex-valued ext-real-valued real-valued set
R + (- r0) is Relation-like NAT -defined Function-like non empty total complex-valued ext-real-valued real-valued set
lim (R - r0) is complex real ext-real Element of REAL
(R . 0) - (lim r0) is complex real ext-real Element of REAL
r0 - R is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
- R is Relation-like NAT -defined Function-like non empty total complex-valued ext-real-valued real-valued set
(- 1) (#) R is Relation-like NAT -defined Function-like non empty total complex-valued ext-real-valued real-valued set
r0 + (- R) is Relation-like NAT -defined Function-like non empty total complex-valued ext-real-valued real-valued set
lim (r0 - R) is complex real ext-real Element of REAL
(lim r0) - (R . 0) is complex real ext-real Element of REAL
R (#) r0 is Relation-like NAT -defined REAL -valued Function-like non empty total V18( NAT , REAL ) complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
lim (R (#) r0) is complex real ext-real Element of REAL
(R . 0) * (lim r0) is complex real ext-real Element of REAL
lim R is complex real ext-real Element of REAL
(lim R) + (lim r0) is complex real ext-real Element of REAL
(lim R) - (lim r0) is complex real ext-real Element of REAL
(lim r0) - (lim R) is complex real ext-real Element of REAL
(lim R) * (lim r0) is complex real ext-real Element of REAL
R is non empty complex-membered ext-real-membered real-membered set
(R) is complex real ext-real set
r1 is complex real ext-real set
r1 - (R) is complex real ext-real Element of REAL
t is ext-real set
(R) + (r1 - (R)) is complex real ext-real Element of REAL
t is complex real ext-real set
r0 is non empty complex-membered ext-real-membered real-membered set
R is complex real ext-real set
(r0) is complex real ext-real set
R is non empty complex-membered ext-real-membered real-membered set
(R) is complex real ext-real set
t is complex real ext-real set
(R) - t is complex real ext-real Element of REAL
m is ext-real set
(R) - ((R) - t) is complex real ext-real Element of REAL
m is complex real ext-real set
R is non empty complex-membered ext-real-membered real-membered set
r0 is complex real ext-real set
(R) is complex real ext-real set
R is non empty complex-membered ext-real-membered real-membered set
(R) is complex real ext-real set
r0 is complex-membered ext-real-membered real-membered set
(r0) is complex real ext-real set
r1 is complex real ext-real set
R is non empty complex-membered ext-real-membered real-membered set
(R) is complex real ext-real set
r0 is complex-membered ext-real-membered real-membered set
(r0) is complex real ext-real set
r1 is complex real ext-real set
R is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
inf R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() finite cardinal set
0c is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of COMPLEX
R is complex Element of COMPLEX
compcomplex . R is complex Element of COMPLEX
addcomplex . (R,(compcomplex . R)) is complex Element of COMPLEX
addcomplex . ((compcomplex . R),R) is complex Element of COMPLEX
R + (compcomplex . R) is complex Element of COMPLEX
- R is complex Element of COMPLEX
R + (- R) is complex Element of COMPLEX
(compcomplex . R) + R is complex Element of COMPLEX
(- R) + R is complex Element of COMPLEX
the_inverseOp_wrt addcomplex is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
id COMPLEX is Relation-like COMPLEX -defined COMPLEX -valued Function-like one-to-one non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
addcomplex * ((id COMPLEX),compcomplex) is Relation-like [:COMPLEX,COMPLEX:] -defined COMPLEX -valued Function-like non empty total V18([:COMPLEX,COMPLEX:], COMPLEX ) complex-valued Element of bool [:[:COMPLEX,COMPLEX:],COMPLEX:]
R is Relation-like [:COMPLEX,COMPLEX:] -defined COMPLEX -valued Function-like non empty total V18([:COMPLEX,COMPLEX:], COMPLEX ) complex-valued Element of bool [:[:COMPLEX,COMPLEX:],COMPLEX:]
r0 is complex Element of COMPLEX
r1 is complex Element of COMPLEX
diffcomplex . (r0,r1) is complex Element of COMPLEX
r0 - r1 is complex Element of COMPLEX
- r1 is complex Element of COMPLEX
r0 + (- r1) is complex Element of COMPLEX
addcomplex . (r0,(- r1)) is complex Element of COMPLEX
compcomplex . r1 is complex Element of COMPLEX
addcomplex . (r0,(compcomplex . r1)) is complex Element of COMPLEX
(addcomplex * ((id COMPLEX),compcomplex)) . (r0,r1) is complex Element of COMPLEX
1r is complex Element of COMPLEX
R is complex Element of COMPLEX
r0 is complex Element of COMPLEX
r1 is complex Element of COMPLEX
addcomplex . (r0,r1) is complex Element of COMPLEX
multcomplex . (R,(addcomplex . (r0,r1))) is complex Element of COMPLEX
r0 + r1 is complex Element of COMPLEX
multcomplex . (R,(r0 + r1)) is complex Element of COMPLEX
R * (r0 + r1) is complex Element of COMPLEX
R * r0 is complex Element of COMPLEX
R * r1 is complex Element of COMPLEX
(R * r0) + (R * r1) is complex Element of COMPLEX
addcomplex . ((R * r0),(R * r1)) is complex Element of COMPLEX
multcomplex . (R,r0) is complex Element of COMPLEX
addcomplex . ((multcomplex . (R,r0)),(R * r1)) is complex Element of COMPLEX
multcomplex . (R,r1) is complex Element of COMPLEX
addcomplex . ((multcomplex . (R,r0)),(multcomplex . (R,r1))) is complex Element of COMPLEX
addcomplex . (R,r0) is complex Element of COMPLEX
multcomplex . ((addcomplex . (R,r0)),r1) is complex Element of COMPLEX
R + r0 is complex Element of COMPLEX
multcomplex . ((R + r0),r1) is complex Element of COMPLEX
(R + r0) * r1 is complex Element of COMPLEX
r0 * r1 is complex Element of COMPLEX
(R * r1) + (r0 * r1) is complex Element of COMPLEX
addcomplex . ((R * r1),(r0 * r1)) is complex Element of COMPLEX
addcomplex . ((multcomplex . (R,r1)),(r0 * r1)) is complex Element of COMPLEX
multcomplex . (r0,r1) is complex Element of COMPLEX
addcomplex . ((multcomplex . (R,r1)),(multcomplex . (r0,r1))) is complex Element of COMPLEX
R is complex set
multcomplex [;] (R,(id COMPLEX)) is Relation-like Function-like set
r0 is complex Element of COMPLEX
multcomplex [;] (r0,(id COMPLEX)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
R is complex Element of COMPLEX
multcomplex [;] (R,(id COMPLEX)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
r0 is complex Element of COMPLEX
(multcomplex [;] (R,(id COMPLEX))) . r0 is complex Element of COMPLEX
R * r0 is complex Element of COMPLEX
(id COMPLEX) . r0 is complex Element of COMPLEX
multcomplex . (R,((id COMPLEX) . r0)) is complex Element of COMPLEX
multcomplex . (R,r0) is complex Element of COMPLEX
R is complex Element of COMPLEX
(R) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (R,(id COMPLEX)) is Relation-like Function-like set
r0 is complex Element of COMPLEX
(R) . r0 is complex Element of COMPLEX
R * r0 is complex Element of COMPLEX
R is complex Element of COMPLEX
(R) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (R,(id COMPLEX)) is Relation-like Function-like set
[:COMPLEX,REAL:] is Relation-like non empty non trivial complex-valued ext-real-valued real-valued non finite set
bool [:COMPLEX,REAL:] is non empty non trivial non finite set
() is Relation-like COMPLEX -defined REAL -valued Function-like non empty total V18( COMPLEX , REAL ) complex-valued ext-real-valued real-valued Element of bool [:COMPLEX,REAL:]
R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R + r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
R + r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of bool [:NAT,COMPLEX:]
[:NAT,COMPLEX:] is Relation-like non empty non trivial complex-valued non finite set
bool [:NAT,COMPLEX:] is non empty non trivial non finite set
rng (R + r0) is complex-membered finite Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
addcomplex .: (R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
dom addcomplex is Relation-like COMPLEX -defined COMPLEX -valued non empty complex-valued Element of bool [:COMPLEX,COMPLEX:]
rng R is complex-membered finite Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
rng r0 is complex-membered finite Element of bool COMPLEX
[:(rng R),(rng r0):] is Relation-like complex-valued finite set
R + r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of bool [:NAT,COMPLEX:]
[:NAT,COMPLEX:] is Relation-like non empty non trivial complex-valued non finite set
bool [:NAT,COMPLEX:] is non empty non trivial non finite set
dom (R + r0) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
(dom R) /\ (dom r0) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom (addcomplex .: (R,r0)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
t is set
(R + r0) . t is complex set
R . t is complex set
r0 . t is complex set
(R . t) + (r0 . t) is complex Element of COMPLEX
addcomplex . ((R . t),(r0 . t)) is set
(addcomplex .: (R,r0)) . t is complex set
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R - r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
- r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
- 1 is complex real ext-real non positive set
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
R + (- r0) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
R - r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of bool [:NAT,COMPLEX:]
[:NAT,COMPLEX:] is Relation-like non empty non trivial complex-valued non finite set
bool [:NAT,COMPLEX:] is non empty non trivial non finite set
rng (R - r0) is complex-membered finite Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
diffcomplex .: (R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
dom diffcomplex is Relation-like COMPLEX -defined COMPLEX -valued non empty complex-valued Element of bool [:COMPLEX,COMPLEX:]
rng R is complex-membered finite Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
rng r0 is complex-membered finite Element of bool COMPLEX
[:(rng R),(rng r0):] is Relation-like complex-valued finite set
dom (diffcomplex .: (R,r0)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
(dom R) /\ (dom r0) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
R - r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of bool [:NAT,COMPLEX:]
[:NAT,COMPLEX:] is Relation-like non empty non trivial complex-valued non finite set
bool [:NAT,COMPLEX:] is non empty non trivial non finite set
dom (R - r0) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
t is set
(R - r0) . t is complex set
R . t is complex set
r0 . t is complex set
(R . t) - (r0 . t) is complex Element of COMPLEX
diffcomplex . ((R . t),(r0 . t)) is set
(diffcomplex .: (R,r0)) . t is complex set
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
- R is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) R is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
- R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of bool [:NAT,COMPLEX:]
[:NAT,COMPLEX:] is Relation-like non empty non trivial complex-valued non finite set
bool [:NAT,COMPLEX:] is non empty non trivial non finite set
rng (- R) is complex-membered finite Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
compcomplex * R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
dom compcomplex is non empty complex-membered Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
rng R is complex-membered finite Element of bool COMPLEX
dom (compcomplex * R) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
- R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of bool [:NAT,COMPLEX:]
[:NAT,COMPLEX:] is Relation-like non empty non trivial complex-valued non finite set
bool [:NAT,COMPLEX:] is non empty non trivial non finite set
dom (- R) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
r1 is set
(- R) . r1 is complex set
R . r1 is complex set
- (R . r1) is complex Element of COMPLEX
compcomplex . (R . r1) is complex set
(compcomplex * R) . r1 is complex set
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 is complex set
r0 (#) R is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
r0 (#) R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of bool [:NAT,COMPLEX:]
[:NAT,COMPLEX:] is Relation-like non empty non trivial complex-valued non finite set
bool [:NAT,COMPLEX:] is non empty non trivial non finite set
rng (r0 (#) R) is complex-membered finite Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
(r0) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r0,(id COMPLEX)) is Relation-like Function-like set
(r0) * R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
dom (r0) is non empty complex-membered Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
rng R is complex-membered finite Element of bool COMPLEX
r0 (#) R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of bool [:NAT,COMPLEX:]
[:NAT,COMPLEX:] is Relation-like non empty non trivial complex-valued non finite set
bool [:NAT,COMPLEX:] is non empty non trivial non finite set
dom (r0 (#) R) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom ((r0) * R) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
t is set
R . t is complex set
(r0 (#) R) . t is complex set
r0 * (R . t) is complex Element of COMPLEX
(r0) . (R . t) is complex set
((r0) * R) . t is complex set
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
|.R.| is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
abs R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like Element of bool [:NAT,REAL:]
rng (abs R) is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
() * R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
dom () is non empty complex-membered Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
rng R is complex-membered finite Element of bool COMPLEX
abs R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like Element of bool [:NAT,REAL:]
dom (abs R) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom (() * R) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
r1 is set
R . r1 is complex set
(abs R) . r1 is complex real ext-real Element of REAL
abs (R . r1) is complex real ext-real Element of REAL
Re (R . r1) is complex real ext-real Element of REAL
(Re (R . r1)) ^2 is complex real ext-real Element of REAL
(Re (R . r1)) * (Re (R . r1)) is complex real ext-real set
Im (R . r1) is complex real ext-real Element of REAL
(Im (R . r1)) ^2 is complex real ext-real Element of REAL
(Im (R . r1)) * (Im (R . r1)) is complex real ext-real set
((Re (R . r1)) ^2) + ((Im (R . r1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (R . r1)) ^2) + ((Im (R . r1)) ^2)) is complex real ext-real Element of REAL
() . (R . r1) is complex real ext-real Element of REAL
(() * R) . r1 is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
t is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
COMPLEX * is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
dom r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
len r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
r0 -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = r0 } is set
Seg r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded r0 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= r0 ) } is set
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
r1 . R is complex set
len r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom r1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r0 + r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
r0 -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = r0 } is set
Seg r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded r0 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= r0 ) } is set
r1 is complex Element of COMPLEX
t is complex Element of COMPLEX
r1 + t is complex Element of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
t . R is complex set
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
m . R is complex set
(r0,t,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
addcomplex .: (t,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(r0,t,m) . R is complex set
dom (r0,t,m) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
addcomplex . (r1,t) is complex Element of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
{0c} is functional non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element set
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r1) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
r1 -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = r1 } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R,r0,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r1)
(r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r1)
r1 |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite r1 -element FinSequence-like FinSubsequence-like Element of r1 -tuples_on COMPLEX
Seg r1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded r1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= r1 ) } is set
(Seg r1) --> 0c is Relation-like Seg r1 -defined Seg r1 -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg r1,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg r1),{0c}:]
[:(Seg r1),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg r1),{0c}:] is non empty finite V61() set
(r1,(r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r1)
addcomplex .: ((r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] ((- 1),(id COMPLEX)) is Relation-like Function-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
r0 -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = r0 } is set
Seg r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded r0 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= r0 ) } is set
r1 is complex Element of COMPLEX
- r1 is complex Element of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
t . R is complex set
(r0,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(r0,t) . R is complex set
dom (r0,t) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
compcomplex . r1 is complex Element of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) (R) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex . 0c is complex Element of COMPLEX
R |-> (compcomplex . 0c) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
(Seg R) --> (compcomplex . 0c) is Relation-like Seg R -defined Seg R -defined COMPLEX -valued {(compcomplex . 0c)} -valued Function-like constant total total V18( Seg R,{(compcomplex . 0c)}) complex-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{(compcomplex . 0c)}:]
{(compcomplex . 0c)} is non empty trivial complex-membered finite 1 -element set
[:(Seg R),{(compcomplex . 0c)}:] is Relation-like complex-valued finite set
bool [:(Seg R),{(compcomplex . 0c)}:] is non empty finite V61() set
- 0c is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of COMPLEX
R |-> (- 0c) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
(Seg R) --> (- 0c) is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {(- 0c)} -valued Function-like constant total total V18( Seg R,{(- 0c)}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{(- 0c)}:]
{(- 0c)} is functional non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element set
[:(Seg R),{(- 0c)}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{(- 0c)}:] is non empty finite V61() set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r1) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
r1 -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = r1 } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r1)
(r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r1)
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(r1,(r1,t),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r1)
addcomplex .: ((r1,t),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r1)
r1 |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite r1 -element FinSequence-like FinSubsequence-like Element of r1 -tuples_on COMPLEX
Seg r1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded r1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= r1 ) } is set
(Seg r1) --> 0c is Relation-like Seg r1 -defined Seg r1 -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg r1,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg r1),{0c}:]
[:(Seg r1),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg r1),{0c}:] is non empty finite V61() set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) (R,r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r1,(R,r1))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r1,(R,r1))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,t,r1),(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,t,r1),(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,t,(R,r1,(R,r1))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (t,(R,r1,(R,r1))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R,t,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (t,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) (R,r0,r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0),(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0),(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,r1),(R,(R,r0),(R,r1))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,r1),(R,(R,r0),(R,r1))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r1,r0),(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r1,r0),(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,(R,r1,r0),(R,r0)),(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,(R,r1,r0),(R,r0)),(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,(R,r0,(R,r0))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,(R,r0,(R,r0))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r1,(R,r0,(R,r0))),(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r1,(R,r0,(R,r0))),(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R,r1,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r1,(R)),(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r1,(R)),(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r0 - r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
r0 -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = r0 } is set
Seg r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded r0 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= r0 ) } is set
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
r1 . R is complex set
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
(r0,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
- t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(r0,r1,t) . R is complex set
t . R is complex set
(r1 . R) - (t . R) is complex Element of COMPLEX
dom (r0,r1,t) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
diffcomplex . ((r1 . R),(t . R)) is set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- (R) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) (R) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- (R)) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R),r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R) + (- r0) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: ((R),r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R),(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R),(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- (R,r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) (R,r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- (R,r1)) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) (R,r0,r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 + (- r0) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) (R,r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0),(R,(R,r1))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0),(R,(R,r1))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) (R,r0,r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) (R,r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0),(R,(R,r1))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0),(R,(R,r1))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r0) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) (R,r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R,r0,r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,r1) + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: ((R,r0,r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r1,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- (R,r1,t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) (R,r1,t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- (R,r1,t)) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,(R,r1,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R,r0,(R,r1)),(R,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,(R,r1)),(R,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r1),(R,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r1),(R,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,(R,r1),(R,t))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,(R,r1),(R,t))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r1,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r1,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: ((R,r0,r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,(R,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,(R,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r1,(R,t))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r1,(R,t))) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,r1),(R,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,r1),(R,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r1,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- (R,r1,t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) (R,r1,t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- (R,r1,t)) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,(R,r1,t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R,r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,(R,r1),t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,(R,r1),t)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,(R,r1)),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,(R,r1)),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R,r0,r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,r1),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,t),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,t) + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: ((R,r0,t),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,r1),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,r1) + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: ((R,r0,r1),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R,r0,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r1,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r1,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 + (- r0) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r1,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r1,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,(R,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r1,(R,r0)),r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r1,(R,r0)),r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0),r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0),r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,(R,(R,r0),r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,(R,(R,r0),r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R,r1,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,r1),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,r1),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R,r0,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R,r1),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r1),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,(R,r1),r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,(R,r1),r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,(R,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0,(R,r1)),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,(R,r1)),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is complex Element of COMPLEX
r1 (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(r1) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r1,(id COMPLEX)) is Relation-like Function-like set
(r1) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
r0 -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = r0 } is set
Seg r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded r0 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= r0 ) } is set
r1 is complex Element of COMPLEX
t is complex Element of COMPLEX
t * r1 is complex Element of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
t . R is complex set
(r0,t,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
(t) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (t,(id COMPLEX)) is Relation-like Function-like set
(t) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(r0,t,t) . R is complex set
dom (r0,t,t) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
(t) . r1 is complex Element of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is complex Element of COMPLEX
r1 is complex Element of COMPLEX
r0 * r1 is complex Element of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r1) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r1,(id COMPLEX)) is Relation-like Function-like set
(r1) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,t,r1),r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r0,(id COMPLEX)) is Relation-like Function-like set
(r0) * (R,t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,t,(r0 * r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
((r0 * r1)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] ((r0 * r1),(id COMPLEX)) is Relation-like Function-like set
((r0 * r1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
multcomplex . (r0,r1) is complex Element of COMPLEX
multcomplex [;] ((multcomplex . (r0,r1)),(id COMPLEX)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
(multcomplex [;] ((multcomplex . (r0,r1)),(id COMPLEX))) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
multcomplex [;] (r1,(id COMPLEX)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r0,(multcomplex [;] (r1,(id COMPLEX)))) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
(multcomplex [;] (r0,(multcomplex [;] (r1,(id COMPLEX))))) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
multcomplex [;] (r0,(id COMPLEX)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
(multcomplex [;] (r0,(id COMPLEX))) * (multcomplex [;] (r1,(id COMPLEX))) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
((multcomplex [;] (r0,(id COMPLEX))) * (multcomplex [;] (r1,(id COMPLEX)))) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is complex Element of COMPLEX
r1 is complex Element of COMPLEX
r0 + r1 is complex Element of COMPLEX
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,(r0 + r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
((r0 + r1)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] ((r0 + r1),(id COMPLEX)) is Relation-like Function-like set
((r0 + r1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,t,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r0,(id COMPLEX)) is Relation-like Function-like set
(r0) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r1) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r1,(id COMPLEX)) is Relation-like Function-like set
(r1) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,t,r0),(R,t,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,t,r0),(R,t,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
multcomplex [;] (r0,(id COMPLEX)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r1,(id COMPLEX)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
addcomplex . (r0,r1) is complex Element of COMPLEX
multcomplex [;] ((addcomplex . (r0,r1)),(id COMPLEX)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
(multcomplex [;] ((addcomplex . (r0,r1)),(id COMPLEX))) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
addcomplex .: ((multcomplex [;] (r0,(id COMPLEX))),(multcomplex [;] (r1,(id COMPLEX)))) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
(addcomplex .: ((multcomplex [;] (r0,(id COMPLEX))),(multcomplex [;] (r1,(id COMPLEX))))) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 is complex Element of COMPLEX
(R,(R,r1,t),r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r0,(id COMPLEX)) is Relation-like Function-like set
(r0) * (R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,t,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r1,r0),(R,t,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r1,r0),(R,t,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,1r) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(1r) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (1r,(id COMPLEX)) is Relation-like Function-like set
(1r) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
rng r0 is complex-membered finite Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
(id COMPLEX) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,0c) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(0c) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (0c,(id COMPLEX)) is Relation-like Function-like set
(0c) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
rng r0 is complex-membered finite Element of bool COMPLEX
bool COMPLEX is non empty non trivial non finite set
(id COMPLEX) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
multcomplex [;] (0c,((id COMPLEX) * r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
multcomplex [;] (0c,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
- 1r is complex Element of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,(- 1r)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
((- 1r)) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] ((- 1r),(id COMPLEX)) is Relation-like Function-like set
((- 1r)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
|.r0.| is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
() * r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
R -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = R } is set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
r0 -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = r0 } is set
Seg r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded r0 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= r0 ) } is set
r1 is complex Element of COMPLEX
|.r1.| is complex real ext-real Element of REAL
Re r1 is complex real ext-real Element of REAL
(Re r1) ^2 is complex real ext-real Element of REAL
(Re r1) * (Re r1) is complex real ext-real set
Im r1 is complex real ext-real Element of REAL
(Im r1) ^2 is complex real ext-real Element of REAL
(Im r1) * (Im r1) is complex real ext-real set
((Re r1) ^2) + ((Im r1) ^2) is complex real ext-real Element of REAL
sqrt (((Re r1) ^2) + ((Im r1) ^2)) is complex real ext-real Element of REAL
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (r0)
t . R is complex set
(r0,t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r0 -element FinSequence-like FinSubsequence-like Element of r0 -tuples_on REAL
r0 -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = r0 } is set
() * t is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r0,t) . R is complex real ext-real Element of REAL
len (r0,t) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom (r0,t) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded r0 -element Element of bool NAT
() . r1 is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R,(R)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
R -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = R } is set
() * (R) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
R |-> 0 is Relation-like empty-yielding NAT -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on NAT
R -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b1 where b1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b1 = R } is set
(Seg R) --> 0 is Relation-like Seg R -defined Seg R -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant total total V18( Seg R,{0}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0}:]
{0} is functional non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element set
[:(Seg R),{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0}:] is non empty finite V61() set
() . 0c is complex real ext-real Element of REAL
R |-> (() . 0c) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(Seg R) --> (() . 0c) is Relation-like Seg R -defined Seg R -defined REAL -valued {(() . 0c)} -valued Function-like constant total total V18( Seg R,{(() . 0c)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{(() . 0c)}:]
{(() . 0c)} is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element set
[:(Seg R),{(() . 0c)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg R),{(() . 0c)}:] is non empty finite V61() set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
R -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = R } is set
() * (R,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(R,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
() * r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
r0 . r1 is complex set
(R,r0) . r1 is complex set
(R,(R,r0)) . r1 is complex real ext-real Element of REAL
t is complex Element of COMPLEX
|.t.| is complex real ext-real Element of REAL
Re t is complex real ext-real Element of REAL
(Re t) ^2 is complex real ext-real Element of REAL
(Re t) * (Re t) is complex real ext-real set
Im t is complex real ext-real Element of REAL
(Im t) ^2 is complex real ext-real Element of REAL
(Im t) * (Im t) is complex real ext-real set
((Re t) ^2) + ((Im t) ^2) is complex real ext-real Element of REAL
sqrt (((Re t) ^2) + ((Im t) ^2)) is complex real ext-real Element of REAL
t is complex Element of COMPLEX
- t is complex Element of COMPLEX
|.(- t).| is complex real ext-real Element of REAL
Re (- t) is complex real ext-real Element of REAL
(Re (- t)) ^2 is complex real ext-real Element of REAL
(Re (- t)) * (Re (- t)) is complex real ext-real set
Im (- t) is complex real ext-real Element of REAL
(Im (- t)) ^2 is complex real ext-real Element of REAL
(Im (- t)) * (Im (- t)) is complex real ext-real set
((Re (- t)) ^2) + ((Im (- t)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (- t)) ^2) + ((Im (- t)) ^2)) is complex real ext-real Element of REAL
|.t.| is complex real ext-real Element of REAL
Re t is complex real ext-real Element of REAL
(Re t) ^2 is complex real ext-real Element of REAL
(Re t) * (Re t) is complex real ext-real set
Im t is complex real ext-real Element of REAL
(Im t) ^2 is complex real ext-real Element of REAL
(Im t) * (Im t) is complex real ext-real set
((Re t) ^2) + ((Im t) ^2) is complex real ext-real Element of REAL
sqrt (((Re t) ^2) + ((Im t) ^2)) is complex real ext-real Element of REAL
(R,r0) . r1 is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is complex Element of COMPLEX
|.r0.| is complex real ext-real Element of REAL
Re r0 is complex real ext-real Element of REAL
(Re r0) ^2 is complex real ext-real Element of REAL
(Re r0) * (Re r0) is complex real ext-real set
Im r0 is complex real ext-real Element of REAL
(Im r0) ^2 is complex real ext-real Element of REAL
(Im r0) * (Im r0) is complex real ext-real set
((Re r0) ^2) + ((Im r0) ^2) is complex real ext-real Element of REAL
sqrt (((Re r0) ^2) + ((Im r0) ^2)) is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r0,(id COMPLEX)) is Relation-like Function-like set
(r0) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,r1,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
R -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = R } is set
() * (R,r1,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(R,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
() * r1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
|.r0.| * (R,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
r1 . t is complex set
(R,r1,r0) . t is complex set
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R,r1) . t is complex real ext-real Element of REAL
(R,(R,r1,r0)) . t is complex real ext-real Element of REAL
m is complex Element of COMPLEX
|.m.| is complex real ext-real Element of REAL
Re m is complex real ext-real Element of REAL
(Re m) ^2 is complex real ext-real Element of REAL
(Re m) * (Re m) is complex real ext-real set
Im m is complex real ext-real Element of REAL
(Im m) ^2 is complex real ext-real Element of REAL
(Im m) * (Im m) is complex real ext-real set
((Re m) ^2) + ((Im m) ^2) is complex real ext-real Element of REAL
sqrt (((Re m) ^2) + ((Im m) ^2)) is complex real ext-real Element of REAL
m is complex Element of COMPLEX
r0 * m is complex Element of COMPLEX
|.(r0 * m).| is complex real ext-real Element of REAL
Re (r0 * m) is complex real ext-real Element of REAL
(Re (r0 * m)) ^2 is complex real ext-real Element of REAL
(Re (r0 * m)) * (Re (r0 * m)) is complex real ext-real set
Im (r0 * m) is complex real ext-real Element of REAL
(Im (r0 * m)) ^2 is complex real ext-real Element of REAL
(Im (r0 * m)) * (Im (r0 * m)) is complex real ext-real set
((Re (r0 * m)) ^2) + ((Im (r0 * m)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (r0 * m)) ^2) + ((Im (r0 * m)) ^2)) is complex real ext-real Element of REAL
|.m.| is complex real ext-real Element of REAL
Re m is complex real ext-real Element of REAL
(Re m) ^2 is complex real ext-real Element of REAL
(Re m) * (Re m) is complex real ext-real set
Im m is complex real ext-real Element of REAL
(Im m) ^2 is complex real ext-real Element of REAL
(Im m) * (Im m) is complex real ext-real set
((Re m) ^2) + ((Im m) ^2) is complex real ext-real Element of REAL
sqrt (((Re m) ^2) + ((Im m) ^2)) is complex real ext-real Element of REAL
|.r0.| * |.m.| is complex real ext-real Element of REAL
f1 is complex real ext-real Element of REAL
|.r0.| * f1 is complex real ext-real Element of REAL
(|.r0.| * (R,r1)) . t is complex real ext-real Element of REAL
R is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (R) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(R) (#) (R) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (R)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (R))) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
((R)) is complex real ext-real Element of REAL
((R)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R)) (#) ((R)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R)))) is complex real ext-real Element of REAL
R |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
R -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = R } is set
(Seg R) --> 0 is Relation-like Seg R -defined Seg R -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant total total V18( Seg R,{0}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0}:]
{0} is functional non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element set
[:(Seg R),{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0}:] is non empty finite V61() set
sqr (R |-> 0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(R |-> 0) (#) (R |-> 0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (R |-> 0)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (R |-> 0))) is complex real ext-real Element of REAL
0 ^2 is complex real ext-real Element of REAL
0 * 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
R |-> (0 ^2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(Seg R) --> (0 ^2) is Relation-like Seg R -defined Seg R -defined REAL -valued {(0 ^2)} -valued Function-like constant total total V18( Seg R,{(0 ^2)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{(0 ^2)}:]
{(0 ^2)} is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element set
[:(Seg R),{(0 ^2)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg R),{(0 ^2)}:] is non empty finite V61() set
Sum (R |-> (0 ^2)) is complex real ext-real Element of REAL
sqrt (Sum (R |-> (0 ^2))) is complex real ext-real Element of REAL
R * 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of NAT
sqrt (R * 0) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is complex real ext-real Element of REAL
(r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r0) (#) (r0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r0)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r0))) is complex real ext-real Element of REAL
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
r0 . r1 is complex set
(R,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
R -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = R } is set
sqr (R,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(R,r0) (#) (R,r0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (R,r0)) is complex real ext-real Element of REAL
(R,r0) . r1 is complex real ext-real Element of REAL
R |-> 0 is Relation-like empty-yielding NAT -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on NAT
R -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b1 where b1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b1 = R } is set
(Seg R) --> 0 is Relation-like Seg R -defined Seg R -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant total total V18( Seg R,{0}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0}:]
{0} is functional non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element set
[:(Seg R),{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0}:] is non empty finite V61() set
(R |-> 0) . r1 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of NAT
t is complex Element of COMPLEX
|.t.| is complex real ext-real Element of REAL
Re t is complex real ext-real Element of REAL
(Re t) ^2 is complex real ext-real Element of REAL
(Re t) * (Re t) is complex real ext-real set
Im t is complex real ext-real Element of REAL
(Im t) ^2 is complex real ext-real Element of REAL
(Im t) * (Im t) is complex real ext-real set
((Re t) ^2) + ((Im t) ^2) is complex real ext-real Element of REAL
sqrt (((Re t) ^2) + ((Im t) ^2)) is complex real ext-real Element of REAL
(R |-> 0c) . r1 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is complex real ext-real Element of REAL
(r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r0) (#) (r0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r0)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r0))) is complex real ext-real Element of REAL
(R,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
R -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = R } is set
sqr (R,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(R,r0) (#) (R,r0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (R,r0)) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0)) is complex real ext-real Element of REAL
((R,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0)) (#) ((R,r0)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0)))) is complex real ext-real Element of REAL
(r0) is complex real ext-real Element of REAL
(r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r0) (#) (r0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r0)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r0))) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is complex Element of COMPLEX
|.r0.| is complex real ext-real Element of REAL
Re r0 is complex real ext-real Element of REAL
(Re r0) ^2 is complex real ext-real Element of REAL
(Re r0) * (Re r0) is complex real ext-real set
Im r0 is complex real ext-real Element of REAL
(Im r0) ^2 is complex real ext-real Element of REAL
(Im r0) * (Im r0) is complex real ext-real set
((Re r0) ^2) + ((Im r0) ^2) is complex real ext-real Element of REAL
sqrt (((Re r0) ^2) + ((Im r0) ^2)) is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total V18( COMPLEX , COMPLEX ) complex-valued Element of bool [:COMPLEX,COMPLEX:]
multcomplex [;] (r0,(id COMPLEX)) is Relation-like Function-like set
(r0) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r1,r0)) is complex real ext-real Element of REAL
((R,r1,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r1,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r1,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r1,r0)) (#) ((R,r1,r0)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r1,r0))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r1,r0)))) is complex real ext-real Element of REAL
(r1) is complex real ext-real Element of REAL
(r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r1) (#) (r1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r1)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r1))) is complex real ext-real Element of REAL
|.r0.| * (r1) is complex real ext-real Element of REAL
|.r0.| ^2 is complex real ext-real Element of REAL
|.r0.| * |.r0.| is complex real ext-real set
(R,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
R -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = R } is set
sqr (R,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(R,r1) (#) (R,r1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (R,r1)) is complex real ext-real Element of REAL
|.r0.| * (R,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
sqr (|.r0.| * (R,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(|.r0.| * (R,r1)) (#) (|.r0.| * (R,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (|.r0.| * (R,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr (|.r0.| * (R,r1)))) is complex real ext-real Element of REAL
(|.r0.| ^2) * (sqr (R,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
Sum ((|.r0.| ^2) * (sqr (R,r1))) is complex real ext-real Element of REAL
sqrt (Sum ((|.r0.| ^2) * (sqr (R,r1)))) is complex real ext-real Element of REAL
(|.r0.| ^2) * (Sum (sqr (R,r1))) is complex real ext-real Element of REAL
sqrt ((|.r0.| ^2) * (Sum (sqr (R,r1)))) is complex real ext-real Element of REAL
sqrt (|.r0.| ^2) is complex real ext-real Element of REAL
sqrt (Sum (sqr (R,r1))) is complex real ext-real Element of REAL
(sqrt (|.r0.| ^2)) * (sqrt (Sum (sqr (R,r1)))) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is complex real ext-real Element of REAL
(r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r0) (#) (r0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r0)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r0))) is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) is complex real ext-real Element of REAL
((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,r1)) (#) ((R,r0,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,r1)))) is complex real ext-real Element of REAL
(r1) is complex real ext-real Element of REAL
(r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r1) (#) (r1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r1)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r1))) is complex real ext-real Element of REAL
(r0) + (r1) is complex real ext-real Element of REAL
(R,(R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
R -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = R } is set
sqr (R,(R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(R,(R,r0,r1)) (#) (R,(R,r0,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (R,(R,r0,r1))) is complex real ext-real Element of REAL
(R,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
sqr (R,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(R,r0) (#) (R,r0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (R,r0)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (R,r0))) is complex real ext-real Element of REAL
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(R,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
mlt ((R,r0),(R,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
(mlt ((R,r0),(R,r1))) . t is complex real ext-real Element of REAL
r0 . t is complex set
r1 . t is complex set
(R,r0) . t is complex real ext-real Element of REAL
m is complex Element of COMPLEX
|.m.| is complex real ext-real Element of REAL
Re m is complex real ext-real Element of REAL
(Re m) ^2 is complex real ext-real Element of REAL
(Re m) * (Re m) is complex real ext-real set
Im m is complex real ext-real Element of REAL
(Im m) ^2 is complex real ext-real Element of REAL
(Im m) * (Im m) is complex real ext-real set
((Re m) ^2) + ((Im m) ^2) is complex real ext-real Element of REAL
sqrt (((Re m) ^2) + ((Im m) ^2)) is complex real ext-real Element of REAL
(R,r1) . t is complex real ext-real Element of REAL
m is complex Element of COMPLEX
|.m.| is complex real ext-real Element of REAL
Re m is complex real ext-real Element of REAL
(Re m) ^2 is complex real ext-real Element of REAL
(Re m) * (Re m) is complex real ext-real set
Im m is complex real ext-real Element of REAL
(Im m) ^2 is complex real ext-real Element of REAL
(Im m) * (Im m) is complex real ext-real set
((Re m) ^2) + ((Im m) ^2) is complex real ext-real Element of REAL
sqrt (((Re m) ^2) + ((Im m) ^2)) is complex real ext-real Element of REAL
|.m.| * |.m.| is complex real ext-real Element of REAL
Sum (mlt ((R,r0),(R,r1))) is complex real ext-real Element of REAL
(Sum (mlt ((R,r0),(R,r1)))) ^2 is complex real ext-real Element of REAL
(Sum (mlt ((R,r0),(R,r1)))) * (Sum (mlt ((R,r0),(R,r1)))) is complex real ext-real set
sqrt ((Sum (mlt ((R,r0),(R,r1)))) ^2) is complex real ext-real Element of REAL
sqr (R,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(R,r1) (#) (R,r1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (R,r1)) is complex real ext-real Element of REAL
(Sum (sqr (R,r0))) * (Sum (sqr (R,r1))) is complex real ext-real Element of REAL
sqrt ((Sum (sqr (R,r0))) * (Sum (sqr (R,r1)))) is complex real ext-real Element of REAL
len (mlt ((R,r0),(R,r1))) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom (mlt ((R,r0),(R,r1))) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
2 * (Sum (mlt ((R,r0),(R,r1)))) is complex real ext-real Element of REAL
2 * (sqrt ((Sum (sqr (R,r0))) * (Sum (sqr (R,r1))))) is complex real ext-real Element of REAL
(Sum (sqr (R,r0))) + (2 * (Sum (mlt ((R,r0),(R,r1))))) is complex real ext-real Element of REAL
(Sum (sqr (R,r0))) + (2 * (sqrt ((Sum (sqr (R,r0))) * (Sum (sqr (R,r1)))))) is complex real ext-real Element of REAL
((Sum (sqr (R,r0))) + (2 * (Sum (mlt ((R,r0),(R,r1)))))) + (Sum (sqr (R,r1))) is complex real ext-real Element of REAL
((Sum (sqr (R,r0))) + (2 * (sqrt ((Sum (sqr (R,r0))) * (Sum (sqr (R,r1))))))) + (Sum (sqr (R,r1))) is complex real ext-real Element of REAL
(R,r0) + (R,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
sqr ((R,r0) + (R,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
((R,r0) + (R,r1)) (#) ((R,r0) + (R,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
(sqr (R,(R,r0,r1))) . t is complex real ext-real Element of REAL
(sqr ((R,r0) + (R,r1))) . t is complex real ext-real Element of REAL
len ((R,r0) + (R,r1)) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom ((R,r0) + (R,r1)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
(R,r0,r1) . t is complex set
(R,(R,r0,r1)) . t is complex real ext-real Element of REAL
m is complex Element of COMPLEX
|.m.| is complex real ext-real Element of REAL
Re m is complex real ext-real Element of REAL
(Re m) ^2 is complex real ext-real Element of REAL
(Re m) * (Re m) is complex real ext-real set
Im m is complex real ext-real Element of REAL
(Im m) ^2 is complex real ext-real Element of REAL
(Im m) * (Im m) is complex real ext-real set
((Re m) ^2) + ((Im m) ^2) is complex real ext-real Element of REAL
sqrt (((Re m) ^2) + ((Im m) ^2)) is complex real ext-real Element of REAL
m is complex real ext-real Element of REAL
(R,r0) . t is complex real ext-real Element of REAL
(R,r1) . t is complex real ext-real Element of REAL
r0 . t is complex set
r1 . t is complex set
((R,r0) + (R,r1)) . t is complex real ext-real Element of REAL
f1 is complex Element of COMPLEX
f2 is complex Element of COMPLEX
f1 + f2 is complex Element of COMPLEX
|.(f1 + f2).| is complex real ext-real Element of REAL
Re (f1 + f2) is complex real ext-real Element of REAL
(Re (f1 + f2)) ^2 is complex real ext-real Element of REAL
(Re (f1 + f2)) * (Re (f1 + f2)) is complex real ext-real set
Im (f1 + f2) is complex real ext-real Element of REAL
(Im (f1 + f2)) ^2 is complex real ext-real Element of REAL
(Im (f1 + f2)) * (Im (f1 + f2)) is complex real ext-real set
((Re (f1 + f2)) ^2) + ((Im (f1 + f2)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (f1 + f2)) ^2) + ((Im (f1 + f2)) ^2)) is complex real ext-real Element of REAL
|.f1.| is complex real ext-real Element of REAL
Re f1 is complex real ext-real Element of REAL
(Re f1) ^2 is complex real ext-real Element of REAL
(Re f1) * (Re f1) is complex real ext-real set
Im f1 is complex real ext-real Element of REAL
(Im f1) ^2 is complex real ext-real Element of REAL
(Im f1) * (Im f1) is complex real ext-real set
((Re f1) ^2) + ((Im f1) ^2) is complex real ext-real Element of REAL
sqrt (((Re f1) ^2) + ((Im f1) ^2)) is complex real ext-real Element of REAL
|.f2.| is complex real ext-real Element of REAL
Re f2 is complex real ext-real Element of REAL
(Re f2) ^2 is complex real ext-real Element of REAL
(Re f2) * (Re f2) is complex real ext-real set
Im f2 is complex real ext-real Element of REAL
(Im f2) ^2 is complex real ext-real Element of REAL
(Im f2) * (Im f2) is complex real ext-real set
((Re f2) ^2) + ((Im f2) ^2) is complex real ext-real Element of REAL
sqrt (((Re f2) ^2) + ((Im f2) ^2)) is complex real ext-real Element of REAL
|.f1.| + |.f2.| is complex real ext-real Element of REAL
f2 is complex real ext-real Element of REAL
|.f1.| + f2 is complex real ext-real Element of REAL
f1 is complex real ext-real Element of REAL
f1 + f2 is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
m ^2 is complex real ext-real Element of REAL
m * m is complex real ext-real set
k ^2 is complex real ext-real Element of REAL
k * k is complex real ext-real set
sqrt (Sum (sqr (R,r1))) is complex real ext-real Element of REAL
(sqrt (Sum (sqr (R,r0)))) ^2 is complex real ext-real Element of REAL
(sqrt (Sum (sqr (R,r0)))) * (sqrt (Sum (sqr (R,r0)))) is complex real ext-real set
(sqrt (Sum (sqr (R,r1)))) ^2 is complex real ext-real Element of REAL
(sqrt (Sum (sqr (R,r1)))) * (sqrt (Sum (sqr (R,r1)))) is complex real ext-real set
Sum (sqr ((R,r0) + (R,r1))) is complex real ext-real Element of REAL
2 * (mlt ((R,r0),(R,r1))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
(sqr (R,r0)) + (2 * (mlt ((R,r0),(R,r1)))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
((sqr (R,r0)) + (2 * (mlt ((R,r0),(R,r1))))) + (sqr (R,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on REAL
Sum (((sqr (R,r0)) + (2 * (mlt ((R,r0),(R,r1))))) + (sqr (R,r1))) is complex real ext-real Element of REAL
Sum ((sqr (R,r0)) + (2 * (mlt ((R,r0),(R,r1))))) is complex real ext-real Element of REAL
(Sum ((sqr (R,r0)) + (2 * (mlt ((R,r0),(R,r1)))))) + (Sum (sqr (R,r1))) is complex real ext-real Element of REAL
Sum (2 * (mlt ((R,r0),(R,r1)))) is complex real ext-real Element of REAL
(Sum (sqr (R,r0))) + (Sum (2 * (mlt ((R,r0),(R,r1))))) is complex real ext-real Element of REAL
((Sum (sqr (R,r0))) + (Sum (2 * (mlt ((R,r0),(R,r1)))))) + (Sum (sqr (R,r1))) is complex real ext-real Element of REAL
(sqrt (Sum (sqr (R,r0)))) * (sqrt (Sum (sqr (R,r1)))) is complex real ext-real Element of REAL
2 * ((sqrt (Sum (sqr (R,r0)))) * (sqrt (Sum (sqr (R,r1))))) is complex real ext-real Element of REAL
(Sum (sqr (R,r0))) + (2 * ((sqrt (Sum (sqr (R,r0)))) * (sqrt (Sum (sqr (R,r1)))))) is complex real ext-real Element of REAL
((Sum (sqr (R,r0))) + (2 * ((sqrt (Sum (sqr (R,r0)))) * (sqrt (Sum (sqr (R,r1))))))) + (Sum (sqr (R,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr (R,(R,r0,r1)))) is complex real ext-real Element of REAL
(sqrt (Sum (sqr (R,r0)))) + (sqrt (Sum (sqr (R,r1)))) is complex real ext-real Element of REAL
((sqrt (Sum (sqr (R,r0)))) + (sqrt (Sum (sqr (R,r1))))) ^2 is complex real ext-real Element of REAL
((sqrt (Sum (sqr (R,r0)))) + (sqrt (Sum (sqr (R,r1))))) * ((sqrt (Sum (sqr (R,r0)))) + (sqrt (Sum (sqr (R,r1))))) is complex real ext-real set
sqrt (((sqrt (Sum (sqr (R,r0)))) + (sqrt (Sum (sqr (R,r1))))) ^2) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is complex real ext-real Element of REAL
(r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r0) (#) (r0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r0)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r0))) is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) is complex real ext-real Element of REAL
((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,r1)) (#) ((R,r0,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,r1)))) is complex real ext-real Element of REAL
(r1) is complex real ext-real Element of REAL
(r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r1) (#) (r1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r1)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r1))) is complex real ext-real Element of REAL
(r0) + (r1) is complex real ext-real Element of REAL
(R,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
((R,r1)) is complex real ext-real Element of REAL
((R,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r1)) (#) ((R,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r1)))) is complex real ext-real Element of REAL
(r0) + ((R,r1)) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is complex real ext-real Element of REAL
(r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r0) (#) (r0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r0)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r0))) is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r1) is complex real ext-real Element of REAL
(r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r1) (#) (r1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r1)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r1))) is complex real ext-real Element of REAL
(r0) - (r1) is complex real ext-real Element of REAL
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) is complex real ext-real Element of REAL
((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,r1)) (#) ((R,r0,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,r1)))) is complex real ext-real Element of REAL
(R,(R,r0,r1),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,r1) + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: ((R,r0,r1),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) + (r1) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r0) is complex real ext-real Element of REAL
(r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r0) (#) (r0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r0)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r0))) is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(r1) is complex real ext-real Element of REAL
(r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r1) (#) (r1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r1)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r1))) is complex real ext-real Element of REAL
(r0) - (r1) is complex real ext-real Element of REAL
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) is complex real ext-real Element of REAL
((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,r1)) (#) ((R,r0,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,r1)))) is complex real ext-real Element of REAL
(R,(R,r0,r1),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,r1),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) + (r1) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) is complex real ext-real Element of REAL
((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,r1)) (#) ((R,r0,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,r1)))) is complex real ext-real Element of REAL
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
(R) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R |-> 0c is Relation-like empty-yielding NAT -defined COMPLEX -valued Function-like complex-valued finite R -element FinSequence-like FinSubsequence-like Element of R -tuples_on COMPLEX
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
(Seg R) --> 0c is Relation-like Seg R -defined Seg R -defined COMPLEX -valued RAT -valued {0c} -valued Function-like constant total total V18( Seg R,{0c}) Function-yielding V22() complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Element of bool [:(Seg R),{0c}:]
[:(Seg R),{0c}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg R),{0c}:] is non empty finite V61() set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) is complex real ext-real Element of REAL
((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,r1)) (#) ((R,r0,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,r1)))) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) is complex real ext-real Element of REAL
((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,r1)) (#) ((R,r0,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,r1)))) is complex real ext-real Element of REAL
(R,r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 + (- r0) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r1,r0)) is complex real ext-real Element of REAL
((R,r1,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r1,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r1,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r1,r0)) (#) ((R,r1,r0)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r1,r0))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r1,r0)))) is complex real ext-real Element of REAL
(R,(R,r1,r0)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(- 1) (#) (R,r1,r0) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,r1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,(R,r1,r0))) is complex real ext-real Element of REAL
((R,(R,r1,r0))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,(R,r1,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,(R,r1,r0))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,(R,r1,r0))) (#) ((R,(R,r1,r0))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,(R,r1,r0)))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,(R,r1,r0))))) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) is complex real ext-real Element of REAL
((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,r1)) (#) ((R,r0,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,r1)))) is complex real ext-real Element of REAL
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,t)) is complex real ext-real Element of REAL
((R,r0,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,t)) (#) ((R,r0,t)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,t))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,t)))) is complex real ext-real Element of REAL
(R,t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,t,r1)) is complex real ext-real Element of REAL
((R,t,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,t,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,t,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,t,r1)) (#) ((R,t,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,t,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,t,r1)))) is complex real ext-real Element of REAL
((R,r0,t)) + ((R,t,r1)) is complex real ext-real Element of REAL
(R,(R,r0,t),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,t),t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,(R,(R,r0,t),t),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R,r0,t),t) + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: ((R,(R,r0,t),t),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,(R,(R,r0,t),t),r1)) is complex real ext-real Element of REAL
((R,(R,(R,r0,t),t),r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,(R,(R,r0,t),t),r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,(R,(R,r0,t),t),r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,(R,(R,r0,t),t),r1)) (#) ((R,(R,(R,r0,t),t),r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,(R,(R,r0,t),t),r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,(R,(R,r0,t),t),r1)))) is complex real ext-real Element of REAL
(R,(R,r0,t),(R,t,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,t),(R,t,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,(R,r0,t),(R,t,r1))) is complex real ext-real Element of REAL
((R,(R,r0,t),(R,t,r1))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,(R,r0,t),(R,t,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,(R,r0,t),(R,t,r1))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,(R,r0,t),(R,t,r1))) (#) ((R,(R,r0,t),(R,t,r1))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,(R,r0,t),(R,t,r1)))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,(R,r0,t),(R,t,r1))))) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is functional FinSequence-membered Element of bool (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is functional FinSequence-membered Element of bool (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(t) is complex real ext-real Element of REAL
(t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * t is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(t) (#) (t) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (t)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (t))) is complex real ext-real Element of REAL
(R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
bool (bool (R)) is non empty set
r0 is Element of bool (bool (R))
union r0 is set
r1 is functional FinSequence-membered Element of bool (R)
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t is set
m is functional FinSequence-membered Element of bool (R)
m is complex real ext-real Element of REAL
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(f1) is complex real ext-real Element of REAL
(f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * f1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(f1) (#) (f1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (f1)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f1))) is complex real ext-real Element of REAL
(R,t,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (t,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is functional FinSequence-membered Element of bool (R)
r1 is functional FinSequence-membered Element of bool (R)
r0 /\ r1 is functional FinSequence-membered Element of bool (R)
t is functional FinSequence-membered Element of bool (R)
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
m is complex real ext-real Element of REAL
m is complex real ext-real Element of REAL
min (m,m) is complex real ext-real Element of REAL
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(f1) is complex real ext-real Element of REAL
(f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * f1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(f1) (#) (f1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (f1)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f1))) is complex real ext-real Element of REAL
(R,t,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (t,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r1 is complex real ext-real Element of REAL
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : not r1 <= ((R,b1,r0)) } is set
bool (R) is non empty set
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1] } is set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,r0) is functional FinSequence-membered Element of bool (R)
bool (R) is non empty set
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : not r0 <= ((R,b1,t)) } is set
(R,t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (t,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,t,r1)) is complex real ext-real Element of REAL
((R,t,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,t,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,t,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,t,r1)) (#) ((R,t,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,t,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,t,r1)))) is complex real ext-real Element of REAL
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (t,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,t,t)) is complex real ext-real Element of REAL
((R,t,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,t,t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,t,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,t,t)) (#) ((R,t,t)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,t,t))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,t,t)))) is complex real ext-real Element of REAL
(R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r1,t)) is complex real ext-real Element of REAL
((R,r1,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r1,t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r1,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r1,t)) (#) ((R,r1,t)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r1,t))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r1,t)))) is complex real ext-real Element of REAL
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (t,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,t,t)) is complex real ext-real Element of REAL
((R,t,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,t,t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,t,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,t,t)) (#) ((R,t,t)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,t,t))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,t,t)))) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,r0) is functional FinSequence-membered Element of bool (R)
bool (R) is non empty set
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : not r0 <= ((R,b1,r1)) } is set
(R,r1,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r1,r1)) is complex real ext-real Element of REAL
((R,r1,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r1,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r1,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r1,r1)) (#) ((R,r1,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r1,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r1,r1)))) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
r0 is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,r0) is functional FinSequence-membered Element of bool (R)
bool (R) is non empty set
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : not r0 <= ((R,b1,r1)) } is set
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r1,t)) is complex real ext-real Element of REAL
((R,r1,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r1,t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r1,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r1,t)) (#) ((R,r1,t)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r1,t))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r1,t)))) is complex real ext-real Element of REAL
r0 - ((R,r1,t)) is complex real ext-real Element of REAL
t is complex real ext-real Element of REAL
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(m) is complex real ext-real Element of REAL
(m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(m) (#) (m) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (m)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (m))) is complex real ext-real Element of REAL
(R,t,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (t,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t + ((R,r1,t)) is complex real ext-real Element of REAL
(m) + ((R,r1,t)) is complex real ext-real Element of REAL
(R,(R,r1,t),m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,t) + (- m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: ((R,r1,t),m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r1,(R,t,m)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- (R,t,m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) (R,t,m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * (R,t,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * (R,t,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 + (- (R,t,m)) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r1,(R,t,m)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r1,(R,t,m))) is complex real ext-real Element of REAL
((R,r1,(R,t,m))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r1,(R,t,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r1,(R,t,m))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r1,(R,t,m))) (#) ((R,r1,(R,t,m))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r1,(R,t,m)))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r1,(R,t,m))))) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is functional FinSequence-membered Element of bool (R)
{ ((R,r0,b1)) where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : b1 in r1 } is set
{ H1(b1) where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1] } is set
t is complex-membered ext-real-membered real-membered Element of bool REAL
(t) is complex real ext-real Element of REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
(t) is complex real ext-real Element of REAL
{ H1(b1) where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1] } is set
t is complex real ext-real Element of REAL
m is complex real ext-real Element of REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
(t) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r1 is complex real ext-real Element of REAL
r0 is functional FinSequence-membered Element of bool (R)
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : not r1 <= (R,b1,r0) } is set
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1] } is set
R is complex-membered ext-real-membered real-membered Element of bool REAL
(R) is complex real ext-real Element of REAL
r0 is complex real ext-real Element of REAL
r1 is ext-real set
r1 is complex real ext-real set
(R) + r1 is complex real ext-real Element of REAL
t is complex real ext-real set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is functional FinSequence-membered Element of bool (R)
(R,r0,r1) is complex real ext-real Element of REAL
{ H1(b1) where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1] } is set
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) t is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- t) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,t)) is complex real ext-real Element of REAL
((R,r0,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,t)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,t)) (#) ((R,r0,t)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,t))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,t)))) is complex real ext-real Element of REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
m is complex real ext-real Element of REAL
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,m)) is complex real ext-real Element of REAL
((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,m)) (#) ((R,r0,m)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,m))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,m)))) is complex real ext-real Element of REAL
(t) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(r1) is complex real ext-real Element of REAL
(r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * r1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(r1) (#) (r1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (r1)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (r1))) is complex real ext-real Element of REAL
t is functional FinSequence-membered Element of bool (R)
(R,(R,r0,r1),t) is complex real ext-real Element of REAL
(R,r0,t) is complex real ext-real Element of REAL
(R,r0,t) + (r1) is complex real ext-real Element of REAL
{ H1(b1) where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1] } is set
t is complex-membered ext-real-membered real-membered Element of bool REAL
m is ext-real set
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,(R,r0,r1),m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
(R,r0,r1) + (- m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: ((R,r0,r1),m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,(R,r0,r1),m)) is complex real ext-real Element of REAL
((R,(R,r0,r1),m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,(R,r0,r1),m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,(R,r0,r1),m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,(R,r0,r1),m)) (#) ((R,(R,r0,r1),m)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,(R,r0,r1),m))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,(R,r0,r1),m)))) is complex real ext-real Element of REAL
{ H2(b1) where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1] } is set
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(t) is complex real ext-real Element of REAL
f1 is complex real ext-real Element of REAL
m is complex-membered ext-real-membered real-membered Element of bool REAL
f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f2 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f2 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- f2) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,f2)) is complex real ext-real Element of REAL
((R,r0,f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,f2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,f2)) (#) ((R,r0,f2)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,f2))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,f2)))) is complex real ext-real Element of REAL
(R,(R,r0,r1),f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) + (- f2) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: ((R,r0,r1),f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,(R,r0,r1),f2)) is complex real ext-real Element of REAL
((R,(R,r0,r1),f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,(R,r0,r1),f2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,(R,r0,r1),f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,(R,r0,r1),f2)) (#) ((R,(R,r0,r1),f2)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,(R,r0,r1),f2))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,(R,r0,r1),f2)))) is complex real ext-real Element of REAL
(R,(R,r0,f2),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: ((R,r0,f2),r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,(R,r0,f2),r1)) is complex real ext-real Element of REAL
((R,(R,r0,f2),r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,(R,r0,f2),r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,(R,r0,f2),r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,(R,r0,f2),r1)) (#) ((R,(R,r0,f2),r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,(R,r0,f2),r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,(R,r0,f2),r1)))) is complex real ext-real Element of REAL
f1 + (r1) is complex real ext-real Element of REAL
(R,(R,r0,r1),t) - (r1) is complex real ext-real Element of REAL
(R,r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,m)) is complex real ext-real Element of REAL
((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,m)) (#) ((R,r0,m)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,m))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,m)))) is complex real ext-real Element of REAL
(m) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is functional FinSequence-membered Element of bool (R)
(R,r0,r1) is complex real ext-real Element of REAL
{ H1(b1) where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1] } is set
(R,r0,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r0) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r0)) is complex real ext-real Element of REAL
((R,r0,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,r0)) (#) ((R,r0,r0)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,r0))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,r0)))) is complex real ext-real Element of REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
m is complex real ext-real set
t is complex real ext-real set
m is complex real ext-real set
0 + m is complex real ext-real Element of REAL
t is complex real ext-real set
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,m)) is complex real ext-real Element of REAL
((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,m)) (#) ((R,r0,m)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,m))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,m)))) is complex real ext-real Element of REAL
t is ext-real set
(t) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is functional FinSequence-membered Element of bool (R)
(R,r0,r1) is complex real ext-real Element of REAL
{ H1(b1) where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1] } is set
t is complex real ext-real Element of REAL
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t is complex-membered ext-real-membered real-membered Element of bool REAL
m is ext-real set
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- f1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,f1)) is complex real ext-real Element of REAL
((R,r0,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,f1)) (#) ((R,r0,f1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,f1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,f1)))) is complex real ext-real Element of REAL
(R,r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,m)) is complex real ext-real Element of REAL
((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,m)) (#) ((R,r0,m)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,m))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,m)))) is complex real ext-real Element of REAL
(t) is complex real ext-real Element of REAL
0 + t is complex real ext-real Element of REAL
m is complex real ext-real set
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- f1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,f1)) is complex real ext-real Element of REAL
((R,r0,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,f1)) (#) ((R,r0,f1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,f1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,f1)))) is complex real ext-real Element of REAL
(R,f1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
f1 + (- r0) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (f1,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(f2) is complex real ext-real Element of REAL
(f2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * f2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (f2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(f2) (#) (f2) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (f2)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f2))) is complex real ext-real Element of REAL
(R,r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- r1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,r1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,r1)) is complex real ext-real Element of REAL
((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,r1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,r1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,r1)) (#) ((R,r0,r1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,r1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,r1)))) is complex real ext-real Element of REAL
t is functional FinSequence-membered Element of bool (R)
(R,r0,t) is complex real ext-real Element of REAL
(R,r1,t) is complex real ext-real Element of REAL
((R,r0,r1)) + (R,r1,t) is complex real ext-real Element of REAL
(R,r1,(R,r0,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,(R,r0,r1)) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t is functional FinSequence-membered Element of bool (R)
(R,t,r0) is functional FinSequence-membered Element of bool (R)
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : not r0 <= (R,b1,t) } is set
(R,r1,t) is complex real ext-real Element of REAL
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,t) is complex real ext-real Element of REAL
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,t) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t is functional FinSequence-membered Element of bool (R)
(R,t,r0) is functional FinSequence-membered Element of bool (R)
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : not r0 <= (R,b1,t) } is set
(R,r1,t) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is complex real ext-real Element of REAL
r1 is functional FinSequence-membered Element of bool (R)
(R,r1,r0) is functional FinSequence-membered Element of bool (R)
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : not r0 <= (R,b1,r1) } is set
t is set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is complex real ext-real Element of REAL
r1 is functional FinSequence-membered Element of bool (R)
(R,r1,r0) is functional FinSequence-membered Element of bool (R)
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : not r0 <= (R,b1,r1) } is set
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,r1) is complex real ext-real Element of REAL
r0 - (R,t,r1) is complex real ext-real Element of REAL
t is complex real ext-real Element of REAL
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(m) is complex real ext-real Element of REAL
(m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(m) (#) (m) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (m)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (m))) is complex real ext-real Element of REAL
(R,t,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (t,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t + (R,t,r1) is complex real ext-real Element of REAL
(m) + (R,t,r1) is complex real ext-real Element of REAL
(R,(R,t,m),r1) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is functional FinSequence-membered Element of bool (R)
r1 is functional FinSequence-membered Element of bool (R)
{ ((R,b1,b2)) where b1, b2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : ( b1 in r0 & b2 in r1 ) } is set
{ H1(b1,b2) where b1, b2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1,b2] } is set
t is complex-membered ext-real-membered real-membered Element of bool REAL
(t) is complex real ext-real Element of REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
(t) is complex real ext-real Element of REAL
{ H1(b1,b2) where b1, b2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1,b2] } is set
t is complex real ext-real Element of REAL
m is complex real ext-real Element of REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
(t) is complex real ext-real Element of REAL
R is complex-membered ext-real-membered real-membered Element of bool REAL
r0 is complex-membered ext-real-membered real-membered Element of bool REAL
R ++ r0 is complex-membered ext-real-membered real-membered set
R ++ r0 is ext-real-membered set
R is complex-membered ext-real-membered real-membered Element of bool REAL
r0 is complex-membered ext-real-membered real-membered Element of bool REAL
R ++ r0 is complex-membered ext-real-membered real-membered set
R ++ r0 is ext-real-membered set
r1 is complex real ext-real set
t is complex real ext-real set
t is complex real ext-real set
t is complex real ext-real set
r1 + t is complex real ext-real Element of REAL
t is ext-real set
{ (b1 + b2) where b1, b2 is complex Element of COMPLEX : ( b1 in R & b2 in r0 ) } is set
m is complex Element of COMPLEX
m is complex Element of COMPLEX
m + m is complex Element of COMPLEX
f2 is complex real ext-real set
f1 is complex real ext-real set
R is complex-membered ext-real-membered real-membered Element of bool REAL
r0 is complex-membered ext-real-membered real-membered Element of bool REAL
R ++ r0 is complex-membered ext-real-membered real-membered set
R ++ r0 is ext-real-membered set
((R ++ r0)) is complex real ext-real set
(R) is complex real ext-real Element of REAL
(r0) is complex real ext-real Element of REAL
(R) + (r0) is complex real ext-real Element of REAL
r1 is complex real ext-real set
r1 / 2 is complex real ext-real Element of REAL
(R) + (r1 / 2) is complex real ext-real Element of REAL
t is complex real ext-real set
(r0) + (r1 / 2) is complex real ext-real Element of REAL
t is complex real ext-real set
t + t is complex real ext-real Element of REAL
m is complex real ext-real set
m is complex real ext-real set
((R) + (r1 / 2)) + ((r0) + (r1 / 2)) is complex real ext-real Element of REAL
((R) + (r0)) + r1 is complex real ext-real Element of REAL
r1 is complex real ext-real set
{ (b1 + b2) where b1, b2 is complex Element of COMPLEX : ( b1 in R & b2 in r0 ) } is set
t is complex Element of COMPLEX
t is complex Element of COMPLEX
t + t is complex Element of COMPLEX
m is complex real ext-real set
m is complex real ext-real set
r0 is complex-membered ext-real-membered real-membered Element of bool REAL
R is complex-membered ext-real-membered real-membered Element of bool REAL
(r0) is complex real ext-real Element of REAL
(R) is complex real ext-real Element of REAL
r1 is complex real ext-real Element of REAL
t is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is functional FinSequence-membered Element of bool (R)
r1 is functional FinSequence-membered Element of bool (R)
(R,r0,r1) is complex real ext-real Element of REAL
{ H1(b1,b2) where b1, b2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1,b2] } is set
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
m is complex real ext-real Element of REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,m,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
m + (- f1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (m,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,m,f1)) is complex real ext-real Element of REAL
((R,m,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,m,f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,m,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,m,f1)) (#) ((R,m,f1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,m,f1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,m,f1)))) is complex real ext-real Element of REAL
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(t) is complex real ext-real Element of REAL
(R,t,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
t + (- m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (t,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,t,m)) is complex real ext-real Element of REAL
((R,t,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,t,m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,t,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,t,m)) (#) ((R,t,m)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,t,m))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,t,m)))) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is functional FinSequence-membered Element of bool (R)
r1 is functional FinSequence-membered Element of bool (R)
(R,r0,r1) is complex real ext-real Element of REAL
(R,r1,r0) is complex real ext-real Element of REAL
{ H1(b1,b2) where b1, b2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1,b2] } is set
{ H1(b1,b2) where b1, b2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S2[b1,b2] } is set
m is complex real ext-real Element of REAL
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,m,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
m + (- f1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (m,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,m,f1)) is complex real ext-real Element of REAL
((R,m,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,m,f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,m,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,m,f1)) (#) ((R,m,f1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,m,f1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,m,f1)))) is complex real ext-real Element of REAL
f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,f2,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
f2 + (- f1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (f2,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,f2,f1)) is complex real ext-real Element of REAL
((R,f2,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,f2,f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,f2,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,f2,f1)) (#) ((R,f2,f1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,f2,f1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,f2,f1)))) is complex real ext-real Element of REAL
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,m,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
m + (- f1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (m,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,m,f1)) is complex real ext-real Element of REAL
((R,m,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,m,f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,m,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,m,f1)) (#) ((R,m,f1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,m,f1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,m,f1)))) is complex real ext-real Element of REAL
f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,f2,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
f2 + (- f1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (f2,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,f2,f1)) is complex real ext-real Element of REAL
((R,f2,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,f2,f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,f2,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,f2,f1)) (#) ((R,f2,f1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,f2,f1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,f2,f1)))) is complex real ext-real Element of REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,m,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
m + (- f1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (m,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,m,f1)) is complex real ext-real Element of REAL
((R,m,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,m,f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,m,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,m,f1)) (#) ((R,m,f1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,m,f1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,m,f1)))) is complex real ext-real Element of REAL
f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,f2,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f1 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
f2 + (- f1) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (f2,f1) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,f2,f1)) is complex real ext-real Element of REAL
((R,f2,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,f2,f1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,f2,f1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,f2,f1)) (#) ((R,f2,f1)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,f2,f1))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,f2,f1)))) is complex real ext-real Element of REAL
(t) is complex real ext-real Element of REAL
(t) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
r1 is functional FinSequence-membered Element of bool (R)
(R,r0,r1) is complex real ext-real Element of REAL
t is functional FinSequence-membered Element of bool (R)
(R,r1,t) is complex real ext-real Element of REAL
(R,r0,t) is complex real ext-real Element of REAL
(R,r0,r1) + (R,r0,t) is complex real ext-real Element of REAL
{ H1(b1) where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S1[b1] } is set
{ H1(b1) where b1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S3[b1] } is set
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t is complex-membered ext-real-membered real-membered Element of bool REAL
(R,r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,m)) is complex real ext-real Element of REAL
((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,m)) (#) ((R,r0,m)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,m))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,m)))) is complex real ext-real Element of REAL
f1 is ext-real set
f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f2 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f2 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- f2) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,f2)) is complex real ext-real Element of REAL
((R,r0,f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,f2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,f2)) (#) ((R,r0,f2)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,f2))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,f2)))) is complex real ext-real Element of REAL
m is complex-membered ext-real-membered real-membered Element of bool REAL
(m) is complex real ext-real Element of REAL
(t) is complex real ext-real Element of REAL
(R,r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) m is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- m) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,m)) is complex real ext-real Element of REAL
((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,m)) (#) ((R,r0,m)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,m))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,m)))) is complex real ext-real Element of REAL
{ H2(b1,b2) where b1, b2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R) : S2[b1,b2] } is set
f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
f1 is complex real ext-real Element of REAL
m ++ t is complex-membered ext-real-membered real-membered set
m ++ t is ext-real-membered set
{ (b1 + b2) where b1, b2 is complex Element of COMPLEX : ( b1 in m & b2 in t ) } is set
f2 is complex Element of COMPLEX
k is complex Element of COMPLEX
f2 + k is complex Element of COMPLEX
k is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,k) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- k is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) k is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * k is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * k is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- k) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,k) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,k)) is complex real ext-real Element of REAL
((R,r0,k)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,k) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,k)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,k)) (#) ((R,r0,k)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,k))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,k)))) is complex real ext-real Element of REAL
i is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,i) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- i is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) i is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * i is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * i is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- i) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,i) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,i)) is complex real ext-real Element of REAL
((R,r0,i)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,i) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,i)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,i)) (#) ((R,r0,i)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,i))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,i)))) is complex real ext-real Element of REAL
(R,i,k) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
i + (- k) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (i,k) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,i,k)) is complex real ext-real Element of REAL
((R,i,k)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,i,k) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,i,k)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,i,k)) (#) ((R,i,k)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,i,k))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,i,k)))) is complex real ext-real Element of REAL
(R,i,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) r0 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * r0 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
i + (- r0) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (i,r0) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,i,r0)) is complex real ext-real Element of REAL
((R,i,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,i,r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,i,r0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,i,r0)) (#) ((R,i,r0)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,i,r0))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,i,r0)))) is complex real ext-real Element of REAL
i is complex real ext-real Element of REAL
f1 is complex-membered ext-real-membered real-membered Element of bool REAL
f1 is ext-real set
f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
k is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,f2,k) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- k is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) k is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * k is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * k is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
f2 + (- k) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (f2,k) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,f2,k)) is complex real ext-real Element of REAL
((R,f2,k)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,f2,k) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,f2,k)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,f2,k)) (#) ((R,f2,k)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,f2,k))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,f2,k)))) is complex real ext-real Element of REAL
(R,r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f2 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f2 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- f2) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,f2)) is complex real ext-real Element of REAL
((R,r0,f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,f2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,f2)) (#) ((R,r0,f2)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,f2))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,f2)))) is complex real ext-real Element of REAL
f1 is ext-real set
f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f2 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f2 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- f2) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,f2)) is complex real ext-real Element of REAL
((R,r0,f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,f2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,f2)) (#) ((R,r0,f2)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,f2))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,f2)))) is complex real ext-real Element of REAL
(f1) is complex real ext-real Element of REAL
(R,r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
- f2 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
(- 1) (#) f2 is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
((- 1)) * f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
compcomplex * f2 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r0 + (- f2) is Relation-like NAT -defined Function-like complex-valued finite FinSequence-like FinSubsequence-like set
diffcomplex .: (r0,f2) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
((R,r0,f2)) is complex real ext-real Element of REAL
((R,r0,f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * (R,r0,f2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr ((R,r0,f2)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
((R,r0,f2)) (#) ((R,r0,f2)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr ((R,r0,f2))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((R,r0,f2)))) is complex real ext-real Element of REAL
((R,r0,f2)) + ((R,r0,m)) is complex real ext-real Element of REAL
((m ++ t)) is complex real ext-real set
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is functional FinSequence-membered Element of bool (R)
r1 is functional FinSequence-membered Element of bool (R)
(R,r0,r1) is complex real ext-real Element of REAL
t is set
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(R,t,r0) is complex real ext-real Element of REAL
(R,t,r1) is complex real ext-real Element of REAL
0 + 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() finite finite-yielding V61() bounded_below bounded_above real-bounded V71() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of NAT
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
{ b1 where b1 is functional FinSequence-membered Element of bool (R) : b1 is (R) } is set
bool (bool (R)) is non empty set
r1 is set
t is functional FinSequence-membered Element of bool (R)
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
(R) is Element of bool (bool (R))
bool (bool (R)) is non empty set
{ b1 where b1 is functional FinSequence-membered Element of bool (R) : b1 is (R) } is set
r0 is functional FinSequence-membered Element of bool (R)
r1 is functional FinSequence-membered Element of bool (R)
r1 is functional FinSequence-membered Element of bool (R)
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(R) is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
R -tuples_on COMPLEX is functional non empty FinSequence-membered FinSequenceSet of COMPLEX
{ b1 where b1 is Relation-like NAT -defined COMPLEX -valued Function-like finite FinSequence-like FinSubsequence-like Element of COMPLEX * : len b1 = R } is set
bool (R) is non empty set
r0 is functional FinSequence-membered Element of bool (R)
r0 ` is functional FinSequence-membered Element of bool (R)
(R) \ r0 is set
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t is complex real ext-real Element of REAL
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(t) is complex real ext-real Element of REAL
(t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * t is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(t) (#) (t) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (t)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (t))) is complex real ext-real Element of REAL
(R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
r1 is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
t is complex real ext-real Element of REAL
t is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(t) is complex real ext-real Element of REAL
(t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * t is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (t) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(t) (#) (t) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (t)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (t))) is complex real ext-real Element of REAL
(R,r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,t) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
m is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
(m) is complex real ext-real Element of REAL
(m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
() * m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
sqr (m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(m) (#) (m) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like set
Sum (sqr (m)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (m))) is complex real ext-real Element of REAL
(R,r1,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like Element of (R)
addcomplex .: (r1,m) is Relation-like NAT -defined COMPLEX -valued Function-like complex-valued finite FinSequence-like FinSubsequence-like FinSequence of COMPLEX
R is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
card R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
(R) is complex real ext-real Element of REAL
(R) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
card r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
(r0) is complex real ext-real Element of REAL
(r0) is complex real ext-real Element of REAL
r1 is set
{r1} is non empty trivial finite 1 -element set
t is complex real ext-real Element of REAL
the complex real ext-real Element of r0 is complex real ext-real Element of r0
t is complex real ext-real Element of REAL
(t) is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element Element of bool REAL
r0 \ (t) is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
t is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
(t) is complex real ext-real Element of REAL
max (t,(t)) is complex real ext-real Element of REAL
(t) is complex real ext-real Element of REAL
min (t,(t)) is complex real ext-real Element of REAL
card (r0 \ (t)) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
card (t) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of omega
(card r0) - (card (t)) is complex real ext-real V48() V49() Element of INT
card t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
(R + 1) - 1 is complex real ext-real V48() V49() Element of INT
f2 is complex real ext-real set
f1 is complex real ext-real set
k is complex real ext-real set
(max (t,(t))) - f2 is complex real ext-real Element of REAL
k is complex real ext-real set
(max (t,(t))) - f2 is complex real ext-real Element of REAL
k is complex real ext-real set
(max (t,(t))) - f2 is complex real ext-real Element of REAL
k is complex real ext-real set
(max (t,(t))) - f2 is complex real ext-real Element of REAL
f2 is complex real ext-real set
f1 is complex real ext-real set
k is complex real ext-real set
(min (t,(t))) + f2 is complex real ext-real Element of REAL
k is complex real ext-real set
(min (t,(t))) + f2 is complex real ext-real Element of REAL
k is complex real ext-real set
(min (t,(t))) + f2 is complex real ext-real Element of REAL
k is complex real ext-real set
(min (t,(t))) + f2 is complex real ext-real Element of REAL
f1 is complex real ext-real set
f1 is complex real ext-real set
R is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
(R) is complex real ext-real Element of REAL
(R) is complex real ext-real Element of REAL
card R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
r0 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
card r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
(r0) is complex real ext-real Element of REAL
(r0) is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
R + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
R + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
R + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
(R + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
R is non empty set
r1 is Relation-like NAT -defined R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of R
len r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is Relation-like NAT -defined R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of R
r0 ^ r1 is Relation-like NAT -defined R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of R
len r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t + (len r0) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0 ^ r1) /. (t + (len r0)) is Element of R
r1 /. t is Element of R
len (r0 ^ r1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(len r0) + (len r1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
(r0 ^ r1) . (t + (len r0)) is set
(t + (len r0)) - (len r0) is complex real ext-real V48() V49() Element of INT
r1 . ((t + (len r0)) - (len r0)) is set
R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
dom R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . r0 is complex real ext-real Element of REAL
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . r1 is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
dom R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . r0 is complex real ext-real Element of REAL
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R . r1 is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
Seg r1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded r1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= r1 ) } is set
R | (Seg r1) is Relation-like NAT -defined Seg r1 -defined NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSubsequence-like Element of bool [:NAT,REAL:]
len R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
Seg (len R) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded len R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= len R ) } is set
dom R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
(dom R) /\ (Seg r1) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . t is complex real ext-real Element of REAL
r0 . t is complex real ext-real Element of REAL
R . t is complex real ext-real Element of REAL
R . t is complex real ext-real Element of REAL
len R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
len R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng R is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
card (rng R) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng r0 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
len r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . t is complex real ext-real Element of REAL
r0 . r1 is complex real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng r0 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
card (rng r0) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
r1 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
(r1) is complex real ext-real Element of REAL
((r1)) is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element Element of bool REAL
r1 \ ((r1)) is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
(r1 \ ((r1))) |` r0 is Relation-like NAT -defined REAL -valued r1 \ ((r1)) -valued REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSubsequence-like Element of bool [:NAT,REAL:]
Seq ((r1 \ ((r1))) |` r0) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom ((r1 \ ((r1))) |` r0) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded set
Sgm (dom ((r1 \ ((r1))) |` r0)) is Relation-like NAT -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like FinSequence of NAT
(Sgm (dom ((r1 \ ((r1))) |` r0))) (#) ((r1 \ ((r1))) |` r0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite set
dom ((r1 \ ((r1))) |` r0) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
Sgm (dom ((r1 \ ((r1))) |` r0)) is Relation-like NAT -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like FinSequence of NAT
((r1 \ ((r1))) |` r0) * (Sgm (dom ((r1 \ ((r1))) |` r0))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite Element of bool [:NAT,REAL:]
rng (Sgm (dom ((r1 \ ((r1))) |` r0))) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool REAL
rng (Seq ((r1 \ ((r1))) |` r0)) is finite set
rng ((r1 \ ((r1))) |` r0) is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
f1 is finite set
card f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
card r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
card ((r1)) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of omega
(card r1) - (card ((r1))) is complex real ext-real V48() V49() Element of INT
m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng m is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
card (rng m) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
f2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng f2 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
len f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
<*(r1)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like FinSequence of REAL
[1,(r1)] is set
{1,(r1)} is non empty complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded set
{1} is non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element set
{{1,(r1)},{1}} is non empty finite V61() set
{[1,(r1)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
f2 ^ <*(r1)*> is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
f1 is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng f1 is non empty complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
rng <*(r1)*> is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element Element of bool REAL
f1 \/ (rng <*(r1)*>) is non empty finite set
(rng r0) \ ((r1)) is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
((rng r0) \ ((r1))) \/ ((r1)) is non empty complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
(rng r0) \/ ((r1)) is non empty complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
len f1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
len <*(r1)*> is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
R + (len <*(r1)*>) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom f1 is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
k is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f1 . f2 is complex real ext-real Element of REAL
f1 . k is complex real ext-real Element of REAL
dom f2 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
f2 . f2 is complex real ext-real Element of REAL
dom f2 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
f2 . k is complex real ext-real Element of REAL
f2 . f2 is complex real ext-real Element of REAL
r1 is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng r1 is non empty complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
len r1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng R is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
card (rng R) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
len R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
rng R is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
len r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
rng r0 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
len r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
rng r0 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
r1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
len r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
rng r1 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
t is complex-membered ext-real-membered real-membered Element of bool REAL
(t) is complex real ext-real Element of REAL
dom r1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
r1 . m is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
Seg R is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded R -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R ) } is set
Seg (R + 1) is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded R + 1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= R + 1 ) } is set
(Seg (R + 1)) /\ (Seg R) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
Seg (len r1) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded len r1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= len r1 ) } is set
r1 | (Seg R) is Relation-like NAT -defined Seg R -defined NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSubsequence-like Element of bool [:NAT,REAL:]
dom (r1 | (Seg R)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
dom r0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
Seg (len r0) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded len r0 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT : ( 1 <= b1 & b1 <= len r0 ) } is set
r0 | (Seg R) is Relation-like NAT -defined Seg R -defined NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSubsequence-like Element of bool [:NAT,REAL:]
dom (r0 | (Seg R)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
f2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng f2 is finite set
f1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng f1 is finite set
k is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
r0 . k is complex real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
r0 . (k + 1) is complex real ext-real Element of REAL
i is complex real ext-real Element of REAL
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
r1 . (m + 1) is complex real ext-real Element of REAL
i is complex real ext-real Element of REAL
f1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
len f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
dom f1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
rng f1 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
((t)) is non empty trivial complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded 1 -element Element of bool REAL
(rng r0) \ ((t)) is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
i is set
i is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
f1 . i is complex real ext-real Element of REAL
r0 . i is complex real ext-real Element of REAL
i is set
i is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
r0 . i is complex real ext-real Element of REAL
f1 . i is complex real ext-real Element of REAL
i is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
(len f1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
<*(t)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like FinSequence of REAL
[1,(t)] is set
{1,(t)} is non empty complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded set
{1} is non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element set
{{1,(t)},{1}} is non empty finite V61() set
{[1,(t)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
f1 ^ <*(t)*> is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(f1 ^ <*(t)*>) . i is complex real ext-real Element of REAL
r0 . i is complex real ext-real Element of REAL
<*(t)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like FinSequence of REAL
[1,(t)] is set
{1,(t)} is non empty complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded set
{1} is non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element set
{{1,(t)},{1}} is non empty finite V61() set
{[1,(t)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
f1 ^ <*(t)*> is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(f1 ^ <*(t)*>) . i is complex real ext-real Element of REAL
f1 . i is complex real ext-real Element of REAL
r0 . i is complex real ext-real Element of REAL
<*(t)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like FinSequence of REAL
[1,(t)] is set
{1,(t)} is non empty complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded set
{1} is non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element set
{{1,(t)},{1}} is non empty finite V61() set
{[1,(t)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
f1 ^ <*(t)*> is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(f1 ^ <*(t)*>) . i is complex real ext-real Element of REAL
r0 . i is complex real ext-real Element of REAL
<*(t)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like FinSequence of REAL
[1,(t)] is set
{1,(t)} is non empty complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded set
{1} is non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element set
{{1,(t)},{1}} is non empty finite V61() set
{[1,(t)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
f1 ^ <*(t)*> is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(f1 ^ <*(t)*>) . i is complex real ext-real Element of REAL
r0 . i is complex real ext-real Element of REAL
len (f1 ^ <*(t)*>) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
len <*(t)*> is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
R + (len <*(t)*>) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
f2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
len f2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
f2 ^ <*(t)*> is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
len (f2 ^ <*(t)*>) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
dom f2 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded Element of bool NAT
rng f2 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
(rng r1) \ ((t)) is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
i is set
i is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
f2 . i is complex real ext-real Element of REAL
r1 . i is complex real ext-real Element of REAL
i is set
i is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
r1 . i is complex real ext-real Element of REAL
f2 . i is complex real ext-real Element of REAL
i is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative finite cardinal set
(len f2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of NAT
(f2 ^ <*(t)*>) . i is complex real ext-real Element of REAL
r1 . i is complex real ext-real Element of REAL
(f2 ^ <*(t)*>) . i is complex real ext-real Element of REAL
f2 . i is complex real ext-real Element of REAL
r1 . i is complex real ext-real Element of REAL
(f2 ^ <*(t)*>) . i is complex real ext-real Element of REAL
r1 . i is complex real ext-real Element of REAL
(f2 ^ <*(t)*>) . i is complex real ext-real Element of REAL
r1 . i is complex real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
len R is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
len r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
rng R is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
rng r0 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
R is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng R is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
card (rng R) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of omega
r0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng r0 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
len r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r1 is Relation-like NAT -defined REAL -valued Function-like one-to-one complex-valued ext-real-valued real-valued increasing non-decreasing finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng r1 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
len r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r0 is Relation-like NAT -defined REAL -valued Function-like one-to-one complex-valued ext-real-valued real-valued increasing non-decreasing finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng r0 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
len r0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
r1 is Relation-like NAT -defined REAL -valued Function-like one-to-one complex-valued ext-real-valued real-valued increasing non-decreasing finite FinSequence-like FinSubsequence-like FinSequence of REAL
rng r1 is complex-membered ext-real-membered real-membered finite bounded_below bounded_above real-bounded Element of bool REAL
len r1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
R is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like FinSequence of REAL
(R) is Relation-like NAT -defined REAL -valued Function-like one-to-one complex-valued ext-real-valued real-valued increasing non-decreasing finite FinSequence-like FinSubsequence-like FinSequence of REAL
len (R) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite bounded_below bounded_above real-bounded cardinal Element of NAT
rng R is non empty complex-membered ext-real-membered real-membered finite left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
card (rng R) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite left_end right_end bounded_below bounded_above real-bounded cardinal Element of omega
(0) is functional non empty trivial complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered finite V61() left_end right_end bounded_below bounded_above real-bounded 1 -element Element of bool REAL
R is complex-membered ext-real-membered real-membered Element of bool REAL
r0 is ext-real set
R is non empty complex-membered ext-real-membered real-membered bounded_below Element of bool REAL
r0 is non empty complex-membered ext-real-membered real-membered bounded_below Element of bool REAL
R \/ r0 is non empty complex-membered ext-real-membered real-membered Element of bool REAL
((R \/ r0)) is complex real ext-real Element of REAL
(R) is complex real ext-real Element of REAL
inf R is complex real ext-real set
(r0) is complex real ext-real Element of REAL
inf r0 is complex real ext-real set
min ((R),(r0)) is complex real ext-real Element of REAL
t is complex real ext-real set
t is complex real ext-real set
t is complex real ext-real set
t is complex real ext-real set
R is non empty complex-membered ext-real-membered real-membered bounded_above Element of bool REAL
r0 is non empty complex-membered ext-real-membered real-membered bounded_above Element of bool REAL
R \/ r0 is non empty complex-membered ext-real-membered real-membered Element of bool REAL
((R \/ r0)) is complex real ext-real Element of REAL
(R) is complex real ext-real Element of REAL
sup R is complex real ext-real set
(r0) is complex real ext-real Element of REAL
sup r0 is complex real ext-real set
max ((R),(r0)) is complex real ext-real Element of REAL
t is complex real ext-real set
t is complex real ext-real set
t is complex real ext-real set
t is complex real ext-real set
R is non empty complex-membered ext-real-membered real-membered Element of bool REAL
(R) is complex real ext-real Element of REAL
r0 is complex real ext-real set
r1 is ext-real set
(R) - r0 is complex real ext-real Element of REAL
(R) - ((R) - r0) is complex real ext-real Element of REAL
t is complex real ext-real set