:: SIN_COS semantic presentation

REAL is non empty V46() V47() V48() V52() V65() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V46() V47() V48() V49() V50() V51() V52() Element of K19(REAL)
K19(REAL) is set
COMPLEX is non empty V46() V52() V65() set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
{{},1} is non empty V46() V47() V48() V49() V50() V51() set
INT is non empty V46() V47() V48() V49() V50() V52() V65() set
K20(REAL,REAL) is complex-valued ext-real-valued real-valued set
K19(K20(REAL,REAL)) is set
K20(NAT,REAL) is complex-valued ext-real-valued real-valued set
K19(K20(NAT,REAL)) is set
omega is non empty epsilon-transitive epsilon-connected ordinal V46() V47() V48() V49() V50() V51() V52() set
K19(omega) is set
K19(NAT) is set
RAT is non empty V46() V47() V48() V49() V52() V65() set
K20(NAT,COMPLEX) is complex-valued set
K19(K20(NAT,COMPLEX)) is set
K20(COMPLEX,COMPLEX) is complex-valued set
K19(K20(COMPLEX,COMPLEX)) is set
K20(K20(COMPLEX,COMPLEX),COMPLEX) is complex-valued set
K19(K20(K20(COMPLEX,COMPLEX),COMPLEX)) is set
K20(K20(REAL,REAL),REAL) is complex-valued ext-real-valued real-valued set
K19(K20(K20(REAL,REAL),REAL)) is set
K20(RAT,RAT) is RAT -valued complex-valued ext-real-valued real-valued set
K19(K20(RAT,RAT)) is set
K20(K20(RAT,RAT),RAT) is RAT -valued complex-valued ext-real-valued real-valued set
K19(K20(K20(RAT,RAT),RAT)) is set
K20(INT,INT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued set
K19(K20(INT,INT)) is set
K20(K20(INT,INT),INT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued set
K19(K20(K20(INT,INT),INT)) is set
K20(NAT,NAT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K20(K20(NAT,NAT),NAT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K19(K20(K20(NAT,NAT),NAT)) is set
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() V62() V63() Element of NAT
K19(K20(NAT,NAT)) is set
1r is complex Element of COMPLEX
Re 0 is complex real ext-real Element of REAL
Im 0 is complex real ext-real Element of REAL
Re 1r is complex real ext-real Element of REAL
Im 1r is complex real ext-real Element of REAL
<i> is complex Element of COMPLEX
K88(REAL,0,1,0,1) is Relation-like {0,1} -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20({0,1},REAL))
{0,1} is non empty V46() V47() V48() V49() V50() V51() set
K20({0,1},REAL) is complex-valued ext-real-valued real-valued set
K19(K20({0,1},REAL)) is set
1r *' is complex Element of COMPLEX
(Im 1r) * <i> is complex set
(Re 1r) - ((Im 1r) * <i>) is complex set
- ((Im 1r) * <i>) is complex set
(Re 1r) + (- ((Im 1r) * <i>)) is complex set
|.0.| is complex real ext-real V63() 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) ^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
|.1r.| is complex real ext-real Element of REAL
(Re 1r) ^2 is complex real ext-real Element of REAL
(Re 1r) * (Re 1r) is complex real ext-real set
(Im 1r) ^2 is complex real ext-real Element of REAL
(Im 1r) * (Im 1r) is complex real ext-real set
((Re 1r) ^2) + ((Im 1r) ^2) is complex real ext-real Element of REAL
sqrt (((Re 1r) ^2) + ((Im 1r) ^2)) is complex real ext-real Element of REAL
0c is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of COMPLEX
{0} is non empty V46() V47() V48() V49() V50() V51() set
0 ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
1 ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
2 ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
- 1 is non empty complex real ext-real non positive negative set
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq,bq) is complex Element of COMPLEX
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a,bq) is complex real ext-real Element of COMPLEX
F₁(bq,a) is complex set
(a,bq) * F₁(bq,a) is complex set
1 * F₁(bq,a) is complex set
0 * F₁(bq,a) is complex set
b is set
th1 is set
bq is Relation-like Function-like set
dom bq is set
a is set
bq . a is set
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
F₁(bq,b) is complex set
a is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
b is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th1 is complex Element of COMPLEX
F₁(bq,th1) is complex set
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
F₁(bq,th2) is complex set
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a,bq) is complex real ext-real Element of COMPLEX
F₁(bq,a) is complex real ext-real set
(a,bq) * F₁(bq,a) is complex real ext-real set
1 * F₁(bq,a) is complex real ext-real Element of REAL
0 * F₁(bq,a) is complex real ext-real Element of REAL
b is set
th1 is set
bq is Relation-like Function-like set
dom bq is set
a is set
bq . a is set
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
F₁(bq,b) is complex real ext-real set
a is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
b is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th1 is complex real ext-real Element of REAL
F₁(bq,th1) is complex real ext-real set
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
F₁(bq,th2) is complex real ext-real set
bq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
bq . 0 is complex real ext-real Element of REAL
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . (bq + 1) is complex real ext-real Element of REAL
bq . bq is complex real ext-real Element of REAL
(bq . bq) * (bq + 1) is complex real ext-real Element of REAL
bq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
bq . 0 is complex real ext-real Element of REAL
bq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
bq . 0 is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . a is complex real ext-real Element of REAL
bq . a is complex real ext-real Element of REAL
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . (a + 1) is complex real ext-real Element of REAL
bq . (a + 1) is complex real ext-real Element of REAL
(bq . a) * (a + 1) is complex real ext-real Element of REAL
() is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative set
bq ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . bq is complex real ext-real Element of REAL
() . 0 is complex real ext-real Element of REAL
0 ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative set
() . bq is complex real ext-real Element of REAL
bq ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
() . (bq + 1) is complex real ext-real Element of REAL
(() . bq) * (bq + 1) is complex real ext-real Element of REAL
(bq + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
() . 0 is complex real ext-real Element of REAL
() . 1 is complex real ext-real Element of REAL
() . 2 is complex real ext-real Element of REAL
bq is complex set
bq is complex real ext-real set
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . bq is complex real ext-real Element of REAL
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (bq + 1) is complex real ext-real Element of REAL
bq ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . bq is complex real ext-real Element of REAL
(bq !) * (bq + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq -' 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (bq -' 1) is complex real ext-real Element of REAL
((bq -' 1) !) * bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . bq is complex real ext-real Element of REAL
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq -' bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq -' bq) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (bq -' bq) is complex real ext-real Element of REAL
bq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq + 1) - bq is complex real ext-real Element of REAL
- bq is complex real ext-real non positive set
(bq + 1) + (- bq) is complex real ext-real set
((bq -' bq) !) * ((bq + 1) - bq) is complex real ext-real Element of REAL
(bq + 1) -' bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((bq + 1) -' bq) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((bq + 1) -' bq) is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a - 1 is complex real ext-real Element of REAL
a + (- 1) is complex real ext-real set
(a - 1) + 1 is complex real ext-real Element of REAL
a ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . a is complex real ext-real Element of REAL
(a -' 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (a -' 1) is complex real ext-real Element of REAL
((a -' 1) !) * a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a + 1) -' b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a + 1) - b is complex real ext-real Element of REAL
- b is complex real ext-real non positive set
(a + 1) + (- b) is complex real ext-real set
a - b is complex real ext-real Element of REAL
a + (- b) is complex real ext-real set
(a - b) + 1 is complex real ext-real Element of REAL
a -' b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a -' b) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((a + 1) -' b) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((a + 1) -' b) is complex real ext-real Element of REAL
(a -' b) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (a -' b) is complex real ext-real Element of REAL
0 * <i> is complex set
((a -' b) + 1) + (0 * <i>) is complex set
((a -' b) !) * (((a -' b) + 1) + (0 * <i>)) is complex set
((a - b) + 1) + (0 * <i>) is complex set
((a -' b) !) * (((a - b) + 1) + (0 * <i>)) is complex set
((a -' b) !) * ((a + 1) - b) is complex real ext-real Element of REAL
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . bq is complex real ext-real Element of REAL
bq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
a is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . b is complex Element of COMPLEX
b ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . b is complex real ext-real Element of REAL
bq -' b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq -' b) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (bq -' b) is complex real ext-real Element of REAL
(b !) * ((bq -' b) !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
(bq !) / ((b !) * ((bq -' b) !)) is non empty complex real ext-real positive non negative Element of REAL
((b !) * ((bq -' b) !)) " is non empty complex real ext-real positive non negative set
(bq !) * (((b !) * ((bq -' b) !)) ") is non empty complex real ext-real positive non negative set
a . b is complex Element of COMPLEX
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . b is complex Element of COMPLEX
a . b is complex Element of COMPLEX
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
a is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . b is complex Element of COMPLEX
b ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . b is complex real ext-real Element of REAL
bq -' b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq -' b) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (bq -' b) is complex real ext-real Element of REAL
(b !) * ((bq -' b) !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
1r / ((b !) * ((bq -' b) !)) is complex Element of COMPLEX
((b !) * ((bq -' b) !)) " is non empty complex real ext-real positive non negative set
1r * (((b !) * ((bq -' b) !)) ") is complex set
a . b is complex Element of COMPLEX
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . b is complex Element of COMPLEX
a . b is complex Element of COMPLEX
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
bq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
bq . 0 is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . (a + 1) is complex Element of COMPLEX
bq . a is complex Element of COMPLEX
bq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
bq . 0 is complex Element of COMPLEX
a is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
a . 0 is complex Element of COMPLEX
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . b is complex Element of COMPLEX
a . b is complex Element of COMPLEX
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . (b + 1) is complex Element of COMPLEX
a . (b + 1) is complex Element of COMPLEX
bq . b is complex Element of COMPLEX
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
bq is complex Element of COMPLEX
a is complex Element of COMPLEX
b is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th1 is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex Element of COMPLEX
(bq) . th2 is complex Element of COMPLEX
bq |^ th2 is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . th2 is complex Element of COMPLEX
((bq) . th2) * (bq |^ th2) is complex set
bq -' th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a |^ (bq -' th2) is complex set
a GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a GeoSeq) . (bq -' th2) is complex Element of COMPLEX
(((bq) . th2) * (bq |^ th2)) * (a |^ (bq -' th2)) is complex set
th1 . th2 is complex Element of COMPLEX
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex Element of COMPLEX
th1 . th2 is complex Element of COMPLEX
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
bq is complex Element of COMPLEX
a is complex Element of COMPLEX
b is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th1 is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex Element of COMPLEX
(bq) . th2 is complex Element of COMPLEX
bq |^ th2 is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . th2 is complex Element of COMPLEX
((bq) . th2) * (bq |^ th2) is complex set
bq -' th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a |^ (bq -' th2) is complex set
a GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a GeoSeq) . (bq -' th2) is complex Element of COMPLEX
(((bq) . th2) * (bq |^ th2)) * (a |^ (bq -' th2)) is complex set
th1 . th2 is complex Element of COMPLEX
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex Element of COMPLEX
th1 . th2 is complex Element of COMPLEX
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
a is complex Element of COMPLEX
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
b is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th1 is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex Element of COMPLEX
(bq) . th2 is complex Element of COMPLEX
bq -' th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (a)) . (bq -' th2) is complex Element of COMPLEX
((bq) . th2) * ((Partial_Sums (a)) . (bq -' th2)) is complex Element of COMPLEX
th1 . th2 is complex Element of COMPLEX
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex Element of COMPLEX
th1 . th2 is complex Element of COMPLEX
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is complex real ext-real set
(bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
bq is complex real ext-real set
(bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(Partial_Sums (bq)) . a is complex real ext-real Element of REAL
b is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
th1 is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex real ext-real Element of REAL
(bq) . th2 is complex real ext-real Element of REAL
a -' th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . (a -' th2) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) - ((Partial_Sums (bq)) . (a -' th2)) is complex real ext-real Element of REAL
- ((Partial_Sums (bq)) . (a -' th2)) is complex real ext-real set
((Partial_Sums (bq)) . a) + (- ((Partial_Sums (bq)) . (a -' th2))) is complex real ext-real set
((bq) . th2) * (((Partial_Sums (bq)) . a) - ((Partial_Sums (bq)) . (a -' th2))) is complex real ext-real Element of REAL
th1 . th2 is complex real ext-real Element of REAL
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex real ext-real Element of REAL
th1 . th2 is complex real ext-real Element of REAL
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
bq is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (bq)) . a is complex Element of COMPLEX
b is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th1 is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex Element of COMPLEX
(bq) . th2 is complex Element of COMPLEX
a -' th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . (a -' th2) is complex Element of COMPLEX
((Partial_Sums (bq)) . a) - ((Partial_Sums (bq)) . (a -' th2)) is complex Element of COMPLEX
- ((Partial_Sums (bq)) . (a -' th2)) is complex set
((Partial_Sums (bq)) . a) + (- ((Partial_Sums (bq)) . (a -' th2))) is complex set
((bq) . th2) * (((Partial_Sums (bq)) . a) - ((Partial_Sums (bq)) . (a -' th2))) is complex Element of COMPLEX
th1 . th2 is complex Element of COMPLEX
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex Element of COMPLEX
th1 . th2 is complex Element of COMPLEX
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
a * <i> is complex set
bq + (a * <i>) is complex set
bq is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
b * <i> is complex set
bq + (b * <i>) is complex set
(bq + (a * <i>)) * (bq + (b * <i>)) is complex set
bq * bq is complex real ext-real Element of REAL
a * b is complex real ext-real Element of REAL
(bq * bq) - (a * b) is complex real ext-real Element of REAL
- (a * b) is complex real ext-real set
(bq * bq) + (- (a * b)) is complex real ext-real set
bq * b is complex real ext-real Element of REAL
bq * a is complex real ext-real Element of REAL
(bq * b) + (bq * a) is complex real ext-real Element of REAL
((bq * b) + (bq * a)) * <i> is complex set
((bq * bq) - (a * b)) + (((bq * b) + (bq * a)) * <i>) is complex set
(bq + (b * <i>)) *' is complex Element of COMPLEX
Re (bq + (b * <i>)) is complex real ext-real Element of REAL
Im (bq + (b * <i>)) is complex real ext-real Element of REAL
(Im (bq + (b * <i>))) * <i> is complex set
(Re (bq + (b * <i>))) - ((Im (bq + (b * <i>))) * <i>) is complex set
- ((Im (bq + (b * <i>))) * <i>) is complex set
(Re (bq + (b * <i>))) + (- ((Im (bq + (b * <i>))) * <i>)) is complex set
- b is complex real ext-real Element of REAL
(- b) * <i> is complex set
bq + ((- b) * <i>) is complex set
bq - ((Im (bq + (b * <i>))) * <i>) is complex set
bq + (- ((Im (bq + (b * <i>))) * <i>)) is complex set
bq - (b * <i>) is complex set
- (b * <i>) is complex set
bq + (- (b * <i>)) is complex set
0 * <i> is complex set
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq + 1) + (0 * <i>) is complex set
bq is complex set
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq) . (bq + 1) is complex Element of COMPLEX
(bq) . bq is complex Element of COMPLEX
((bq) . bq) * bq is complex set
(((bq) . bq) * bq) / ((bq + 1) + (0 * <i>)) is complex Element of COMPLEX
((bq + 1) + (0 * <i>)) " is complex set
(((bq) . bq) * bq) * (((bq + 1) + (0 * <i>)) ") is complex set
(bq) . 0 is complex Element of COMPLEX
|.((bq) . bq).| is complex real ext-real Element of REAL
Re ((bq) . bq) is complex real ext-real Element of REAL
(Re ((bq) . bq)) ^2 is complex real ext-real Element of REAL
(Re ((bq) . bq)) * (Re ((bq) . bq)) is complex real ext-real set
Im ((bq) . bq) is complex real ext-real Element of REAL
(Im ((bq) . bq)) ^2 is complex real ext-real Element of REAL
(Im ((bq) . bq)) * (Im ((bq) . bq)) is complex real ext-real set
((Re ((bq) . bq)) ^2) + ((Im ((bq) . bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . bq)) ^2) + ((Im ((bq) . bq)) ^2)) is complex real ext-real Element of REAL
|.bq.| is complex real ext-real Element of REAL
Re bq is complex real ext-real Element of REAL
(Re bq) ^2 is complex real ext-real Element of REAL
(Re bq) * (Re bq) is complex real ext-real set
Im bq is complex real ext-real Element of REAL
(Im bq) ^2 is complex real ext-real Element of REAL
(Im bq) * (Im bq) is complex real ext-real set
((Re bq) ^2) + ((Im bq) ^2) is complex real ext-real Element of REAL
sqrt (((Re bq) ^2) + ((Im bq) ^2)) is complex real ext-real Element of REAL
(|.bq.|) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(|.bq.|) . bq is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq) . (a + 1) is complex Element of COMPLEX
bq |^ (a + 1) is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . (a + 1) is complex Element of COMPLEX
(a + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (a + 1) is complex real ext-real Element of REAL
(bq |^ (a + 1)) / ((a + 1) !) is complex Element of COMPLEX
((a + 1) !) " is non empty complex real ext-real positive non negative set
(bq |^ (a + 1)) * (((a + 1) !) ") is complex set
(bq GeoSeq) . a is complex Element of COMPLEX
((bq GeoSeq) . a) * bq is complex set
(((bq GeoSeq) . a) * bq) / ((a + 1) !) is complex Element of COMPLEX
(((bq GeoSeq) . a) * bq) * (((a + 1) !) ") is complex set
bq |^ a is complex set
(bq |^ a) * bq is complex set
a ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . a is complex real ext-real Element of REAL
(a + 1) + (0 * <i>) is complex set
(a !) * ((a + 1) + (0 * <i>)) is complex set
((bq |^ a) * bq) / ((a !) * ((a + 1) + (0 * <i>))) is complex Element of COMPLEX
((a !) * ((a + 1) + (0 * <i>))) " is complex set
((bq |^ a) * bq) * (((a !) * ((a + 1) + (0 * <i>))) ") is complex set
(bq |^ a) / (a !) is complex Element of COMPLEX
(a !) " is non empty complex real ext-real positive non negative set
(bq |^ a) * ((a !) ") is complex set
bq / ((a + 1) + (0 * <i>)) is complex Element of COMPLEX
((a + 1) + (0 * <i>)) " is complex set
bq * (((a + 1) + (0 * <i>)) ") is complex set
((bq |^ a) / (a !)) * (bq / ((a + 1) + (0 * <i>))) is complex Element of COMPLEX
((bq |^ a) / (a !)) * bq is complex set
(((bq |^ a) / (a !)) * bq) / ((a + 1) + (0 * <i>)) is complex Element of COMPLEX
(((bq |^ a) / (a !)) * bq) * (((a + 1) + (0 * <i>)) ") is complex set
(bq) . a is complex Element of COMPLEX
((bq) . a) * bq is complex set
(((bq) . a) * bq) / ((a + 1) + (0 * <i>)) is complex Element of COMPLEX
(((bq) . a) * bq) * (((a + 1) + (0 * <i>)) ") is complex set
bq |^ 0 is complex set
(bq GeoSeq) . 0 is complex Element of COMPLEX
(bq |^ 0) / (0 !) is complex Element of COMPLEX
(0 !) " is non empty complex real ext-real positive non negative set
(bq |^ 0) * ((0 !) ") is complex set
(|.bq.|) . 0 is complex real ext-real Element of REAL
|.bq.| |^ 0 is complex real ext-real Element of REAL
|.bq.| GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(|.bq.| GeoSeq) . 0 is complex Element of COMPLEX
(|.bq.| |^ 0) / (0 !) is complex real ext-real Element of REAL
(|.bq.| |^ 0) * ((0 !) ") is complex real ext-real set
1 / (() . 0) is complex real ext-real Element of REAL
(() . 0) " is complex real ext-real set
1 * ((() . 0) ") is complex real ext-real set
1 / 1 is non empty complex real ext-real positive non negative Element of REAL
1 " is non empty complex real ext-real positive non negative set
1 * (1 ") is non empty complex real ext-real positive non negative set
|.((bq) . 0).| is complex real ext-real Element of REAL
Re ((bq) . 0) is complex real ext-real Element of REAL
(Re ((bq) . 0)) ^2 is complex real ext-real Element of REAL
(Re ((bq) . 0)) * (Re ((bq) . 0)) is complex real ext-real set
Im ((bq) . 0) is complex real ext-real Element of REAL
(Im ((bq) . 0)) ^2 is complex real ext-real Element of REAL
(Im ((bq) . 0)) * (Im ((bq) . 0)) is complex real ext-real set
((Re ((bq) . 0)) ^2) + ((Im ((bq) . 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . 0)) ^2) + ((Im ((bq) . 0)) ^2)) is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq) . a is complex Element of COMPLEX
|.((bq) . a).| is complex real ext-real Element of REAL
Re ((bq) . a) is complex real ext-real Element of REAL
(Re ((bq) . a)) ^2 is complex real ext-real Element of REAL
(Re ((bq) . a)) * (Re ((bq) . a)) is complex real ext-real set
Im ((bq) . a) is complex real ext-real Element of REAL
(Im ((bq) . a)) ^2 is complex real ext-real Element of REAL
(Im ((bq) . a)) * (Im ((bq) . a)) is complex real ext-real set
((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2)) is complex real ext-real Element of REAL
(|.bq.|) . a is complex real ext-real Element of REAL
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a + 1) + (0 * <i>) is complex set
|.((a + 1) + (0 * <i>)).| is complex real ext-real Element of REAL
Re ((a + 1) + (0 * <i>)) is complex real ext-real Element of REAL
(Re ((a + 1) + (0 * <i>))) ^2 is complex real ext-real Element of REAL
(Re ((a + 1) + (0 * <i>))) * (Re ((a + 1) + (0 * <i>))) is complex real ext-real set
Im ((a + 1) + (0 * <i>)) is complex real ext-real Element of REAL
(Im ((a + 1) + (0 * <i>))) ^2 is complex real ext-real Element of REAL
(Im ((a + 1) + (0 * <i>))) * (Im ((a + 1) + (0 * <i>))) is complex real ext-real set
((Re ((a + 1) + (0 * <i>))) ^2) + ((Im ((a + 1) + (0 * <i>))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a + 1) + (0 * <i>))) ^2) + ((Im ((a + 1) + (0 * <i>))) ^2)) is complex real ext-real Element of REAL
(bq) . (a + 1) is complex Element of COMPLEX
|.((bq) . (a + 1)).| is complex real ext-real Element of REAL
Re ((bq) . (a + 1)) is complex real ext-real Element of REAL
(Re ((bq) . (a + 1))) ^2 is complex real ext-real Element of REAL
(Re ((bq) . (a + 1))) * (Re ((bq) . (a + 1))) is complex real ext-real set
Im ((bq) . (a + 1)) is complex real ext-real Element of REAL
(Im ((bq) . (a + 1))) ^2 is complex real ext-real Element of REAL
(Im ((bq) . (a + 1))) * (Im ((bq) . (a + 1))) is complex real ext-real set
((Re ((bq) . (a + 1))) ^2) + ((Im ((bq) . (a + 1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . (a + 1))) ^2) + ((Im ((bq) . (a + 1))) ^2)) is complex real ext-real Element of REAL
((bq) . a) * bq is complex set
(((bq) . a) * bq) / ((a + 1) + (0 * <i>)) is complex Element of COMPLEX
((a + 1) + (0 * <i>)) " is complex set
(((bq) . a) * bq) * (((a + 1) + (0 * <i>)) ") is complex set
|.((((bq) . a) * bq) / ((a + 1) + (0 * <i>))).| is complex real ext-real Element of REAL
Re ((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) is complex real ext-real Element of REAL
(Re ((((bq) . a) * bq) / ((a + 1) + (0 * <i>)))) ^2 is complex real ext-real Element of REAL
(Re ((((bq) . a) * bq) / ((a + 1) + (0 * <i>)))) * (Re ((((bq) . a) * bq) / ((a + 1) + (0 * <i>)))) is complex real ext-real set
Im ((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) is complex real ext-real Element of REAL
(Im ((((bq) . a) * bq) / ((a + 1) + (0 * <i>)))) ^2 is complex real ext-real Element of REAL
(Im ((((bq) . a) * bq) / ((a + 1) + (0 * <i>)))) * (Im ((((bq) . a) * bq) / ((a + 1) + (0 * <i>)))) is complex real ext-real set
((Re ((((bq) . a) * bq) / ((a + 1) + (0 * <i>)))) ^2) + ((Im ((((bq) . a) * bq) / ((a + 1) + (0 * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((((bq) . a) * bq) / ((a + 1) + (0 * <i>)))) ^2) + ((Im ((((bq) . a) * bq) / ((a + 1) + (0 * <i>)))) ^2)) is complex real ext-real Element of REAL
|.(((bq) . a) * bq).| is complex real ext-real Element of REAL
Re (((bq) . a) * bq) is complex real ext-real Element of REAL
(Re (((bq) . a) * bq)) ^2 is complex real ext-real Element of REAL
(Re (((bq) . a) * bq)) * (Re (((bq) . a) * bq)) is complex real ext-real set
Im (((bq) . a) * bq) is complex real ext-real Element of REAL
(Im (((bq) . a) * bq)) ^2 is complex real ext-real Element of REAL
(Im (((bq) . a) * bq)) * (Im (((bq) . a) * bq)) is complex real ext-real set
((Re (((bq) . a) * bq)) ^2) + ((Im (((bq) . a) * bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((bq) . a) * bq)) ^2) + ((Im (((bq) . a) * bq)) ^2)) is complex real ext-real Element of REAL
|.(((bq) . a) * bq).| / |.((a + 1) + (0 * <i>)).| is complex real ext-real Element of REAL
|.((a + 1) + (0 * <i>)).| " is complex real ext-real set
|.(((bq) . a) * bq).| * (|.((a + 1) + (0 * <i>)).| ") is complex real ext-real set
((|.bq.|) . a) * |.bq.| is complex real ext-real Element of REAL
(((|.bq.|) . a) * |.bq.|) / |.((a + 1) + (0 * <i>)).| is complex real ext-real Element of REAL
(((|.bq.|) . a) * |.bq.|) * (|.((a + 1) + (0 * <i>)).| ") is complex real ext-real set
|.bq.| |^ a is complex real ext-real Element of REAL
(|.bq.| GeoSeq) . a is complex Element of COMPLEX
a ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . a is complex real ext-real Element of REAL
(|.bq.| |^ a) / (a !) is complex real ext-real Element of REAL
(a !) " is non empty complex real ext-real positive non negative set
(|.bq.| |^ a) * ((a !) ") is complex real ext-real set
((|.bq.| |^ a) / (a !)) * |.bq.| is complex real ext-real Element of REAL
(((|.bq.| |^ a) / (a !)) * |.bq.|) / |.((a + 1) + (0 * <i>)).| is complex real ext-real Element of REAL
(((|.bq.| |^ a) / (a !)) * |.bq.|) * (|.((a + 1) + (0 * <i>)).| ") is complex real ext-real set
(|.bq.| |^ a) * |.bq.| is complex real ext-real Element of REAL
(a !) * |.((a + 1) + (0 * <i>)).| is complex real ext-real Element of REAL
((|.bq.| |^ a) * |.bq.|) / ((a !) * |.((a + 1) + (0 * <i>)).|) is complex real ext-real Element of REAL
((a !) * |.((a + 1) + (0 * <i>)).|) " is complex real ext-real set
((|.bq.| |^ a) * |.bq.|) * (((a !) * |.((a + 1) + (0 * <i>)).|) ") is complex real ext-real set
(a + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (a + 1) is complex real ext-real Element of REAL
((|.bq.| |^ a) * |.bq.|) / ((a + 1) !) is complex real ext-real Element of REAL
((a + 1) !) " is non empty complex real ext-real positive non negative set
((|.bq.| |^ a) * |.bq.|) * (((a + 1) !) ") is complex real ext-real set
|.bq.| |^ (a + 1) is complex real ext-real Element of REAL
(|.bq.| GeoSeq) . (a + 1) is complex Element of COMPLEX
(|.bq.| |^ (a + 1)) / ((a + 1) !) is complex real ext-real Element of REAL
(|.bq.| |^ (a + 1)) * (((a + 1) !) ") is complex real ext-real set
(|.bq.|) . (a + 1) is complex real ext-real Element of REAL
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq) . bq is complex Element of COMPLEX
bq . (bq -' 1) is complex Element of COMPLEX
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative set
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq - 1 is complex real ext-real Element of REAL
bq + (- 1) is complex real ext-real set
bq . a is complex Element of COMPLEX
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums bq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums bq) . bq is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (bq)) . bq is complex Element of COMPLEX
bq . bq is complex Element of COMPLEX
((Partial_Sums (bq)) . bq) + (bq . bq) is complex Element of COMPLEX
(Partial_Sums bq) . 0 is complex Element of COMPLEX
bq . 0 is complex Element of COMPLEX
0c + (bq . 0) is complex Element of COMPLEX
(bq) . 0 is complex Element of COMPLEX
((bq) . 0) + (bq . 0) is complex Element of COMPLEX
(Partial_Sums (bq)) . 0 is complex Element of COMPLEX
((Partial_Sums (bq)) . 0) + (bq . 0) is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums bq) . a is complex Element of COMPLEX
(Partial_Sums (bq)) . a is complex Element of COMPLEX
bq . a is complex Element of COMPLEX
((Partial_Sums (bq)) . a) + (bq . a) is complex Element of COMPLEX
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums bq) . (a + 1) is complex Element of COMPLEX
(Partial_Sums (bq)) . (a + 1) is complex Element of COMPLEX
bq . (a + 1) is complex Element of COMPLEX
((Partial_Sums (bq)) . (a + 1)) + (bq . (a + 1)) is complex Element of COMPLEX
(((Partial_Sums (bq)) . a) + (bq . a)) + (bq . (a + 1)) is complex Element of COMPLEX
(bq) . (a + 1) is complex Element of COMPLEX
((Partial_Sums (bq)) . a) + ((bq) . (a + 1)) is complex Element of COMPLEX
(((Partial_Sums (bq)) . a) + ((bq) . (a + 1))) + (bq . (a + 1)) is complex Element of COMPLEX
bq is complex Element of COMPLEX
bq is complex Element of COMPLEX
bq + bq is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq + bq) |^ a is complex set
(bq + bq) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq + bq) GeoSeq) . a is complex Element of COMPLEX
(a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (a,bq,bq)) . a is complex Element of COMPLEX
0 -' 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(0,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (0,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (0,bq,bq)) . 0 is complex Element of COMPLEX
(0,bq,bq) . 0 is complex Element of COMPLEX
(0) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(0) . 0 is complex Element of COMPLEX
bq |^ 0 is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . 0 is complex Element of COMPLEX
((0) . 0) * (bq |^ 0) is complex set
bq |^ 0 is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . 0 is complex Element of COMPLEX
(((0) . 0) * (bq |^ 0)) * (bq |^ 0) is complex set
1 * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
1 / (1 * 1) is non empty complex real ext-real positive non negative Element of REAL
(1 * 1) " is non empty complex real ext-real positive non negative set
1 * ((1 * 1) ") is non empty complex real ext-real positive non negative set
(1 / (1 * 1)) * (bq |^ 0) is complex set
((1 / (1 * 1)) * (bq |^ 0)) * (bq |^ 0) is complex set
1r * ((bq GeoSeq) . 0) is complex Element of COMPLEX
(bq + bq) |^ 0 is complex set
((bq + bq) GeoSeq) . 0 is complex Element of COMPLEX
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq + bq) |^ b is complex set
((bq + bq) GeoSeq) . b is complex Element of COMPLEX
(b,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (b,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (b,bq,bq)) . b is complex Element of COMPLEX
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq + bq) |^ (b + 1) is complex set
((bq + bq) GeoSeq) . (b + 1) is complex Element of COMPLEX
((b + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((b + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((b + 1),bq,bq)) . (b + 1) is complex Element of COMPLEX
(((bq + bq) GeoSeq) . b) * (bq + bq) is complex Element of COMPLEX
(bq + bq) (#) (Partial_Sums (b,bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq + bq) (#) (Partial_Sums (b,bq,bq))) . b is complex Element of COMPLEX
(bq + bq) (#) (b,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((bq + bq) (#) (b,bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((bq + bq) (#) (b,bq,bq))) . b is complex Element of COMPLEX
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((bq + bq) (#) (b,bq,bq)) . th1 is complex Element of COMPLEX
(b,bq,bq) . th1 is complex Element of COMPLEX
(bq + bq) * ((b,bq,bq) . th1) is complex Element of COMPLEX
bq * ((b,bq,bq) . th1) is complex Element of COMPLEX
bq * ((b,bq,bq) . th1) is complex Element of COMPLEX
(bq * ((b,bq,bq) . th1)) + (bq * ((b,bq,bq) . th1)) is complex Element of COMPLEX
bq (#) (b,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq (#) (b,bq,bq)) . th1 is complex Element of COMPLEX
((bq (#) (b,bq,bq)) . th1) + (bq * ((b,bq,bq) . th1)) is complex Element of COMPLEX
bq (#) (b,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq (#) (b,bq,bq)) . th1 is complex Element of COMPLEX
((bq (#) (b,bq,bq)) . th1) + ((bq (#) (b,bq,bq)) . th1) is complex Element of COMPLEX
(bq (#) (b,bq,bq)) + (bq (#) (b,bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq (#) (b,bq,bq)) + (bq (#) (b,bq,bq))) . th1 is complex Element of COMPLEX
Partial_Sums (bq (#) (b,bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq (#) (b,bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (bq (#) (b,bq,bq))) + (Partial_Sums (bq (#) (b,bq,bq))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((Partial_Sums (bq (#) (b,bq,bq))) + (Partial_Sums (bq (#) (b,bq,bq)))) . b is complex Element of COMPLEX
(Partial_Sums (bq (#) (b,bq,bq))) . b is complex Element of COMPLEX
(Partial_Sums (bq (#) (b,bq,bq))) . b is complex Element of COMPLEX
((Partial_Sums (bq (#) (b,bq,bq))) . b) + ((Partial_Sums (bq (#) (b,bq,bq))) . b) is complex Element of COMPLEX
(Partial_Sums (bq (#) (b,bq,bq))) . (b + 1) is complex Element of COMPLEX
(bq (#) (b,bq,bq)) . (b + 1) is complex Element of COMPLEX
((Partial_Sums (bq (#) (b,bq,bq))) . b) + ((bq (#) (b,bq,bq)) . (b + 1)) is complex Element of COMPLEX
(b,bq,bq) . (b + 1) is complex Element of COMPLEX
bq * ((b,bq,bq) . (b + 1)) is complex Element of COMPLEX
((Partial_Sums (bq (#) (b,bq,bq))) . b) + (bq * ((b,bq,bq) . (b + 1))) is complex Element of COMPLEX
(Partial_Sums (bq (#) (b,bq,bq))) . (b + 1) is complex Element of COMPLEX
(bq (#) (b,bq,bq)) . (b + 1) is complex Element of COMPLEX
((Partial_Sums (bq (#) (b,bq,bq))) . b) + ((bq (#) (b,bq,bq)) . (b + 1)) is complex Element of COMPLEX
bq * ((b,bq,bq) . (b + 1)) is complex Element of COMPLEX
((Partial_Sums (bq (#) (b,bq,bq))) . b) + (bq * ((b,bq,bq) . (b + 1))) is complex Element of COMPLEX
((bq (#) (b,bq,bq))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((bq (#) (b,bq,bq))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((bq (#) (b,bq,bq)))) . (b + 1) is complex Element of COMPLEX
((Partial_Sums ((bq (#) (b,bq,bq)))) . (b + 1)) + ((bq (#) (b,bq,bq)) . (b + 1)) is complex Element of COMPLEX
0 + b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums ((bq (#) (b,bq,bq)))) . (b + 1)) + 0c is complex Element of COMPLEX
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 - 1 is complex real ext-real Element of REAL
th1 + (- 1) is complex real ext-real set
(b + 1) - 1 is complex real ext-real Element of REAL
(b + 1) + (- 1) is complex real ext-real set
(th1 - 1) + 1 is complex real ext-real Element of REAL
b + 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq (#) (b,bq,bq)) + ((bq (#) (b,bq,bq))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq (#) (b,bq,bq)) + ((bq (#) (b,bq,bq)))) . th1 is complex Element of COMPLEX
(bq (#) (b,bq,bq)) . th1 is complex Element of COMPLEX
((bq (#) (b,bq,bq))) . th1 is complex Element of COMPLEX
((bq (#) (b,bq,bq)) . th1) + (((bq (#) (b,bq,bq))) . th1) is complex Element of COMPLEX
(b,bq,bq) . th1 is complex Element of COMPLEX
bq * ((b,bq,bq) . th1) is complex Element of COMPLEX
(bq * ((b,bq,bq) . th1)) + (((bq (#) (b,bq,bq))) . th1) is complex Element of COMPLEX
(bq (#) (b,bq,bq)) . (th1 -' 1) is complex Element of COMPLEX
(bq * ((b,bq,bq) . th1)) + ((bq (#) (b,bq,bq)) . (th1 -' 1)) is complex Element of COMPLEX
(b,bq,bq) . (th1 -' 1) is complex Element of COMPLEX
bq * ((b,bq,bq) . (th1 -' 1)) is complex Element of COMPLEX
(bq * ((b,bq,bq) . th1)) + (bq * ((b,bq,bq) . (th1 -' 1))) is complex Element of COMPLEX
bq * 0c is complex Element of COMPLEX
(bq * 0c) + (bq * ((b,bq,bq) . (th1 -' 1))) is complex Element of COMPLEX
bq * 0c is complex Element of COMPLEX
0c + (bq * 0c) is complex Element of COMPLEX
((b + 1),bq,bq) . th1 is complex Element of COMPLEX
b -' b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b - b is complex real ext-real Element of REAL
- b is complex real ext-real non positive set
b + (- b) is complex real ext-real set
(b + 1) -' (b + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(b + 1) - (b + 1) is complex real ext-real Element of REAL
- (b + 1) is non empty complex real ext-real non positive negative set
(b + 1) + (- (b + 1)) is complex real ext-real set
(b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(b) . b is complex Element of COMPLEX
b ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . b is complex real ext-real Element of REAL
(b !) * (0 !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
(b !) / ((b !) * (0 !)) is non empty complex real ext-real positive non negative Element of REAL
((b !) * (0 !)) " is non empty complex real ext-real positive non negative set
(b !) * (((b !) * (0 !)) ") is non empty complex real ext-real positive non negative set
((b + 1)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((b + 1)) . (b + 1) is complex Element of COMPLEX
(b + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (b + 1) is complex real ext-real Element of REAL
((b + 1) !) * (0 !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
((b + 1) !) / (((b + 1) !) * (0 !)) is non empty complex real ext-real positive non negative Element of REAL
(((b + 1) !) * (0 !)) " is non empty complex real ext-real positive non negative set
((b + 1) !) * ((((b + 1) !) * (0 !)) ") is non empty complex real ext-real positive non negative set
((bq (#) (b,bq,bq))) . (b + 1) is complex Element of COMPLEX
((bq (#) (b,bq,bq)) . (b + 1)) + (((bq (#) (b,bq,bq))) . (b + 1)) is complex Element of COMPLEX
(bq * ((b,bq,bq) . (b + 1))) + (((bq (#) (b,bq,bq))) . (b + 1)) is complex Element of COMPLEX
(bq * 0c) + (((bq (#) (b,bq,bq))) . (b + 1)) is complex Element of COMPLEX
(bq (#) (b,bq,bq)) . b is complex Element of COMPLEX
(b,bq,bq) . b is complex Element of COMPLEX
bq * ((b,bq,bq) . b) is complex Element of COMPLEX
bq |^ b is complex set
(bq GeoSeq) . b is complex Element of COMPLEX
((b) . b) * (bq |^ b) is complex set
bq |^ (b -' b) is complex set
(bq GeoSeq) . (b -' b) is complex Element of COMPLEX
(((b) . b) * (bq |^ b)) * (bq |^ (b -' b)) is complex set
bq * ((((b) . b) * (bq |^ b)) * (bq |^ (b -' b))) is complex set
((bq GeoSeq) . b) * bq is complex Element of COMPLEX
((b) . b) * (((bq GeoSeq) . b) * bq) is complex Element of COMPLEX
(((b) . b) * (((bq GeoSeq) . b) * bq)) * (bq |^ (b -' b)) is complex set
bq |^ (b + 1) is complex set
(bq GeoSeq) . (b + 1) is complex Element of COMPLEX
(((b + 1)) . (b + 1)) * (bq |^ (b + 1)) is complex set
((((b + 1)) . (b + 1)) * (bq |^ (b + 1))) * (bq |^ (b -' b)) is complex set
b -' 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b - 0 is complex real ext-real non negative Element of REAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
b + (- 0) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative set
(b + 1) -' 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(b + 1) - 0 is non empty complex real ext-real positive non negative Element of REAL
(b + 1) + (- 0) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative set
(b) . 0 is complex Element of COMPLEX
(0 !) * (b !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
(b !) / ((0 !) * (b !)) is non empty complex real ext-real positive non negative Element of REAL
((0 !) * (b !)) " is non empty complex real ext-real positive non negative set
(b !) * (((0 !) * (b !)) ") is non empty complex real ext-real positive non negative set
((b + 1)) . 0 is complex Element of COMPLEX
(0 !) * ((b + 1) !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
((b + 1) !) / ((0 !) * ((b + 1) !)) is non empty complex real ext-real positive non negative Element of REAL
((0 !) * ((b + 1) !)) " is non empty complex real ext-real positive non negative set
((b + 1) !) * (((0 !) * ((b + 1) !)) ") is non empty complex real ext-real positive non negative set
(bq (#) (b,bq,bq)) . 0 is complex Element of COMPLEX
((bq (#) (b,bq,bq))) . 0 is complex Element of COMPLEX
((bq (#) (b,bq,bq)) . 0) + (((bq (#) (b,bq,bq))) . 0) is complex Element of COMPLEX
(b,bq,bq) . 0 is complex Element of COMPLEX
bq * ((b,bq,bq) . 0) is complex Element of COMPLEX
(bq * ((b,bq,bq) . 0)) + (((bq (#) (b,bq,bq))) . 0) is complex Element of COMPLEX
(bq * ((b,bq,bq) . 0)) + 0c is complex Element of COMPLEX
((b) . 0) * (bq |^ 0) is complex set
bq |^ (b -' 0) is complex set
(bq GeoSeq) . (b -' 0) is complex Element of COMPLEX
(((b) . 0) * (bq |^ 0)) * (bq |^ (b -' 0)) is complex set
bq * ((((b) . 0) * (bq |^ 0)) * (bq |^ (b -' 0))) is complex set
(bq GeoSeq) . b is complex Element of COMPLEX
((bq GeoSeq) . b) * bq is complex Element of COMPLEX
(((b) . 0) * (bq |^ 0)) * (((bq GeoSeq) . b) * bq) is complex set
(((b + 1)) . 0) * (bq |^ 0) is complex set
bq |^ ((b + 1) -' 0) is complex set
(bq GeoSeq) . ((b + 1) -' 0) is complex Element of COMPLEX
((((b + 1)) . 0) * (bq |^ 0)) * (bq |^ ((b + 1) -' 0)) is complex set
th1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(th1 + 1) - 1 is complex real ext-real Element of REAL
(th1 + 1) + (- 1) is complex real ext-real set
(b) . th1 is complex Element of COMPLEX
bq |^ th1 is complex set
(bq GeoSeq) . th1 is complex Element of COMPLEX
((b) . th1) * (bq |^ th1) is complex set
b -' th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq |^ (b -' th1) is complex set
(bq GeoSeq) . (b -' th1) is complex Element of COMPLEX
(((b) . th1) * (bq |^ th1)) * (bq |^ (b -' th1)) is complex set
bq * ((((b) . th1) * (bq |^ th1)) * (bq |^ (b -' th1))) is complex set
(bq * ((((b) . th1) * (bq |^ th1)) * (bq |^ (b -' th1)))) + (bq * ((b,bq,bq) . (th1 -' 1))) is complex set
bq * (((b) . th1) * (bq |^ th1)) is complex set
(bq * (((b) . th1) * (bq |^ th1))) * (bq |^ (b -' th1)) is complex set
(b) . (th1 -' 1) is complex Element of COMPLEX
bq |^ (th1 -' 1) is complex set
(bq GeoSeq) . (th1 -' 1) is complex Element of COMPLEX
((b) . (th1 -' 1)) * (bq |^ (th1 -' 1)) is complex set
b -' (th1 -' 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq |^ (b -' (th1 -' 1)) is complex set
(bq GeoSeq) . (b -' (th1 -' 1)) is complex Element of COMPLEX
(((b) . (th1 -' 1)) * (bq |^ (th1 -' 1))) * (bq |^ (b -' (th1 -' 1))) is complex set
bq * ((((b) . (th1 -' 1)) * (bq |^ (th1 -' 1))) * (bq |^ (b -' (th1 -' 1)))) is complex set
((bq * (((b) . th1) * (bq |^ th1))) * (bq |^ (b -' th1))) + (bq * ((((b) . (th1 -' 1)) * (bq |^ (th1 -' 1))) * (bq |^ (b -' (th1 -' 1))))) is complex set
bq * (bq |^ th1) is complex set
(bq * (bq |^ th1)) * (bq |^ (b -' th1)) is complex set
((b) . th1) * ((bq * (bq |^ th1)) * (bq |^ (b -' th1))) is complex set
(bq |^ (th1 -' 1)) * bq is complex set
((bq |^ (th1 -' 1)) * bq) * (bq |^ (b -' (th1 -' 1))) is complex set
((b) . (th1 -' 1)) * (((bq |^ (th1 -' 1)) * bq) * (bq |^ (b -' (th1 -' 1)))) is complex set
(((b) . th1) * ((bq * (bq |^ th1)) * (bq |^ (b -' th1)))) + (((b) . (th1 -' 1)) * (((bq |^ (th1 -' 1)) * bq) * (bq |^ (b -' (th1 -' 1))))) is complex set
(b -' th1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b - th1 is complex real ext-real Element of REAL
- th1 is complex real ext-real non positive set
b + (- th1) is complex real ext-real set
(b - th1) + 1 is complex real ext-real Element of REAL
(b + 1) - th1 is complex real ext-real Element of REAL
(b + 1) + (- th1) is complex real ext-real set
(b + 1) -' th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b - (th1 -' 1) is complex real ext-real Element of REAL
- (th1 -' 1) is complex real ext-real non positive set
b + (- (th1 -' 1)) is complex real ext-real set
b - (th1 - 1) is complex real ext-real Element of REAL
- (th1 - 1) is complex real ext-real set
b + (- (th1 - 1)) is complex real ext-real set
(th1 -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq |^ (b -' th1)) * bq is complex set
bq |^ ((b -' th1) + 1) is complex set
(bq GeoSeq) . ((b -' th1) + 1) is complex Element of COMPLEX
((b) . th1) + ((b) . (th1 -' 1)) is complex Element of COMPLEX
th1 ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . th1 is complex real ext-real Element of REAL
(b -' th1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (b -' th1) is complex real ext-real Element of REAL
(th1 !) * ((b -' th1) !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
(b !) / ((th1 !) * ((b -' th1) !)) is non empty complex real ext-real positive non negative Element of REAL
((th1 !) * ((b -' th1) !)) " is non empty complex real ext-real positive non negative set
(b !) * (((th1 !) * ((b -' th1) !)) ") is non empty complex real ext-real positive non negative set
((b !) / ((th1 !) * ((b -' th1) !))) + ((b) . (th1 -' 1)) is complex set
(th1 -' 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (th1 -' 1) is complex real ext-real Element of REAL
(b -' (th1 -' 1)) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (b -' (th1 -' 1)) is complex real ext-real Element of REAL
((th1 -' 1) !) * ((b -' (th1 -' 1)) !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
(b !) / (((th1 -' 1) !) * ((b -' (th1 -' 1)) !)) is non empty complex real ext-real positive non negative Element of REAL
(((th1 -' 1) !) * ((b -' (th1 -' 1)) !)) " is non empty complex real ext-real positive non negative set
(b !) * ((((th1 -' 1) !) * ((b -' (th1 -' 1)) !)) ") is non empty complex real ext-real positive non negative set
((b !) / ((th1 !) * ((b -' th1) !))) + ((b !) / (((th1 -' 1) !) * ((b -' (th1 -' 1)) !))) is non empty complex real ext-real positive non negative Element of REAL
((b + 1) - th1) + (0 * <i>) is complex set
((th1 !) * ((b -' th1) !)) * (((b + 1) - th1) + (0 * <i>)) is complex set
((b -' th1) !) * (((b + 1) - th1) + (0 * <i>)) is complex set
(th1 !) * (((b -' th1) !) * (((b + 1) - th1) + (0 * <i>))) is complex set
th1 + (0 * <i>) is complex set
(((th1 -' 1) !) * ((b -' (th1 -' 1)) !)) * (th1 + (0 * <i>)) is complex set
(th1 + (0 * <i>)) * ((th1 -' 1) !) is complex set
((th1 + (0 * <i>)) * ((th1 -' 1) !)) * ((b -' (th1 -' 1)) !) is complex set
((b + 1) -' th1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((b + 1) -' th1) is complex real ext-real Element of REAL
(th1 !) * (((b + 1) -' th1) !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
(b !) * (((b + 1) - th1) + (0 * <i>)) is complex set
((b !) * (((b + 1) - th1) + (0 * <i>))) / (((th1 !) * ((b -' th1) !)) * (((b + 1) - th1) + (0 * <i>))) is complex Element of COMPLEX
(((th1 !) * ((b -' th1) !)) * (((b + 1) - th1) + (0 * <i>))) " is complex set
((b !) * (((b + 1) - th1) + (0 * <i>))) * ((((th1 !) * ((b -' th1) !)) * (((b + 1) - th1) + (0 * <i>))) ") is complex set
((b !) * (((b + 1) - th1) + (0 * <i>))) / ((th1 !) * (((b + 1) -' th1) !)) is complex Element of COMPLEX
((th1 !) * (((b + 1) -' th1) !)) " is non empty complex real ext-real positive non negative set
((b !) * (((b + 1) - th1) + (0 * <i>))) * (((th1 !) * (((b + 1) -' th1) !)) ") is complex set
(b !) * (th1 + (0 * <i>)) is complex set
((b !) * (th1 + (0 * <i>))) / ((th1 !) * (((b + 1) -' th1) !)) is complex Element of COMPLEX
((b !) * (th1 + (0 * <i>))) * (((th1 !) * (((b + 1) -' th1) !)) ") is complex set
((b + 1) - th1) + th1 is complex real ext-real Element of REAL
0 + 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() V62() V63() Element of NAT
(0 + 0) * <i> is complex set
(((b + 1) - th1) + th1) + ((0 + 0) * <i>) is complex set
(b !) * ((((b + 1) - th1) + th1) + ((0 + 0) * <i>)) is complex set
((b !) * ((((b + 1) - th1) + th1) + ((0 + 0) * <i>))) / ((th1 !) * (((b + 1) -' th1) !)) is complex Element of COMPLEX
((b !) * ((((b + 1) - th1) + th1) + ((0 + 0) * <i>))) * (((th1 !) * (((b + 1) -' th1) !)) ") is complex set
((b + 1) !) / ((th1 !) * (((b + 1) -' th1) !)) is non empty complex real ext-real positive non negative Element of REAL
((b + 1) !) * (((th1 !) * (((b + 1) -' th1) !)) ") is non empty complex real ext-real positive non negative set
((b + 1)) . th1 is complex Element of COMPLEX
(((b + 1)) . th1) * (bq |^ th1) is complex set
bq |^ ((b + 1) -' th1) is complex set
(bq GeoSeq) . ((b + 1) -' th1) is complex Element of COMPLEX
((((b + 1)) . th1) * (bq |^ th1)) * (bq |^ ((b + 1) -' th1)) is complex set
(Partial_Sums (bq (#) (b,bq,bq))) + (Partial_Sums ((bq (#) (b,bq,bq)))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((Partial_Sums (bq (#) (b,bq,bq))) + (Partial_Sums ((bq (#) (b,bq,bq))))) . (b + 1) is complex Element of COMPLEX
bq is complex Element of COMPLEX
bq is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
a ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . a is complex real ext-real Element of REAL
1r / (a !) is complex Element of COMPLEX
(a !) " is non empty complex real ext-real positive non negative set
1r * ((a !) ") is complex set
(1r / (a !)) (#) (a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a,bq,bq) . b is complex Element of COMPLEX
(1r / (a !)) * 0c is complex Element of COMPLEX
(a,bq,bq) . b is complex Element of COMPLEX
(1r / (a !)) * ((a,bq,bq) . b) is complex Element of COMPLEX
((1r / (a !)) (#) (a,bq,bq)) . b is complex Element of COMPLEX
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a) . b is complex Element of COMPLEX
bq |^ b is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . b is complex Element of COMPLEX
((a) . b) * (bq |^ b) is complex set
a -' b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq |^ (a -' b) is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . (a -' b) is complex Element of COMPLEX
(((a) . b) * (bq |^ b)) * (bq |^ (a -' b)) is complex set
b ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . b is complex real ext-real Element of REAL
(a -' b) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (a -' b) is complex real ext-real Element of REAL
(b !) * ((a -' b) !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
1r / ((b !) * ((a -' b) !)) is complex Element of COMPLEX
((b !) * ((a -' b) !)) " is non empty complex real ext-real positive non negative set
1r * (((b !) * ((a -' b) !)) ") is complex set
(1r / ((b !) * ((a -' b) !))) * (bq |^ b) is complex set
((1r / ((b !) * ((a -' b) !))) * (bq |^ b)) * (bq |^ (a -' b)) is complex set
(a !) * 1r is complex set
((a !) * 1r) / (a !) is complex Element of COMPLEX
((a !) * 1r) * ((a !) ") is complex set
(((a !) * 1r) / (a !)) / ((b !) * ((a -' b) !)) is complex Element of COMPLEX
(((a !) * 1r) / (a !)) * (((b !) * ((a -' b) !)) ") is complex set
(1r / (a !)) * (a !) is complex set
((1r / (a !)) * (a !)) / ((b !) * ((a -' b) !)) is complex Element of COMPLEX
((1r / (a !)) * (a !)) * (((b !) * ((a -' b) !)) ") is complex set
(bq |^ b) * (bq |^ (a -' b)) is complex set
(((1r / (a !)) * (a !)) / ((b !) * ((a -' b) !))) * ((bq |^ b) * (bq |^ (a -' b))) is complex set
(a !) / ((b !) * ((a -' b) !)) is non empty complex real ext-real positive non negative Element of REAL
(a !) * (((b !) * ((a -' b) !)) ") is non empty complex real ext-real positive non negative set
((a !) / ((b !) * ((a -' b) !))) * (bq |^ b) is complex set
(((a !) / ((b !) * ((a -' b) !))) * (bq |^ b)) * (bq |^ (a -' b)) is complex set
(1r / (a !)) * ((((a !) / ((b !) * ((a -' b) !))) * (bq |^ b)) * (bq |^ (a -' b))) is complex set
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a) . b is complex Element of COMPLEX
((a) . b) * (bq |^ b) is complex set
(((a) . b) * (bq |^ b)) * (bq |^ (a -' b)) is complex set
(1r / (a !)) * ((((a) . b) * (bq |^ b)) * (bq |^ (a -' b))) is complex set
bq is complex Element of COMPLEX
bq is complex Element of COMPLEX
bq + bq is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq + bq) |^ a is complex set
(bq + bq) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq + bq) GeoSeq) . a is complex Element of COMPLEX
a ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . a is complex real ext-real Element of REAL
((bq + bq) |^ a) / (a !) is complex Element of COMPLEX
(a !) " is non empty complex real ext-real positive non negative set
((bq + bq) |^ a) * ((a !) ") is complex set
(a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (a,bq,bq)) . a is complex Element of COMPLEX
(a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (a,bq,bq)) . a is complex Element of COMPLEX
1r / (a !) is complex Element of COMPLEX
1r * ((a !) ") is complex set
((Partial_Sums (a,bq,bq)) . a) * (1r / (a !)) is complex Element of COMPLEX
(1r / (a !)) (#) (Partial_Sums (a,bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((1r / (a !)) (#) (Partial_Sums (a,bq,bq))) . a is complex Element of COMPLEX
(1r / (a !)) (#) (a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((1r / (a !)) (#) (a,bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((1r / (a !)) (#) (a,bq,bq))) . a is complex Element of COMPLEX
(0c) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (0c) is complex Element of COMPLEX
Partial_Sums (0c) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (0c)) is complex Element of COMPLEX
|.(0c).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums |.(0c).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued non-decreasing Element of K19(K20(NAT,REAL))
(Partial_Sums |.(0c).|) . 0 is complex real ext-real Element of REAL
|.(0c).| . 0 is complex real ext-real Element of REAL
(0c) . 0 is complex Element of COMPLEX
|.((0c) . 0).| is complex real ext-real Element of REAL
Re ((0c) . 0) is complex real ext-real Element of REAL
(Re ((0c) . 0)) ^2 is complex real ext-real Element of REAL
(Re ((0c) . 0)) * (Re ((0c) . 0)) is complex real ext-real set
Im ((0c) . 0) is complex real ext-real Element of REAL
(Im ((0c) . 0)) ^2 is complex real ext-real Element of REAL
(Im ((0c) . 0)) * (Im ((0c) . 0)) is complex real ext-real set
((Re ((0c) . 0)) ^2) + ((Im ((0c) . 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((0c) . 0)) ^2) + ((Im ((0c) . 0)) ^2)) is complex real ext-real Element of REAL
(Partial_Sums |.(0c).|) . 0 is complex real ext-real Element of REAL
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative set
(Partial_Sums |.(0c).|) . bq is complex real ext-real Element of REAL
bq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums |.(0c).|) . (bq + 1) is complex real ext-real Element of REAL
(Partial_Sums |.(0c).|) . bq is complex real ext-real Element of REAL
(Partial_Sums |.(0c).|) . (bq + 1) is complex real ext-real Element of REAL
|.(0c).| . (bq + 1) is complex real ext-real Element of REAL
1 + (|.(0c).| . (bq + 1)) is complex real ext-real Element of REAL
(0c) . (bq + 1) is complex Element of COMPLEX
|.((0c) . (bq + 1)).| is complex real ext-real Element of REAL
Re ((0c) . (bq + 1)) is complex real ext-real Element of REAL
(Re ((0c) . (bq + 1))) ^2 is complex real ext-real Element of REAL
(Re ((0c) . (bq + 1))) * (Re ((0c) . (bq + 1))) is complex real ext-real set
Im ((0c) . (bq + 1)) is complex real ext-real Element of REAL
(Im ((0c) . (bq + 1))) ^2 is complex real ext-real Element of REAL
(Im ((0c) . (bq + 1))) * (Im ((0c) . (bq + 1))) is complex real ext-real set
((Re ((0c) . (bq + 1))) ^2) + ((Im ((0c) . (bq + 1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((0c) . (bq + 1))) ^2) + ((Im ((0c) . (bq + 1))) ^2)) is complex real ext-real Element of REAL
1 + |.((0c) . (bq + 1)).| is complex real ext-real Element of REAL
(0c) . bq is complex Element of COMPLEX
((0c) . bq) * 0c is complex Element of COMPLEX
(bq + 1) + (0 * <i>) is complex set
(((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)) is complex Element of COMPLEX
((bq + 1) + (0 * <i>)) " is complex set
(((0c) . bq) * 0c) * (((bq + 1) + (0 * <i>)) ") is complex set
|.((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>))).| is complex real ext-real Element of REAL
Re ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>))) is complex real ext-real Element of REAL
(Re ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)))) ^2 is complex real ext-real Element of REAL
(Re ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)))) * (Re ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)))) is complex real ext-real set
Im ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>))) is complex real ext-real Element of REAL
(Im ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)))) ^2 is complex real ext-real Element of REAL
(Im ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)))) * (Im ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)))) is complex real ext-real set
((Re ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)))) ^2) + ((Im ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)))) ^2) + ((Im ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)))) ^2)) is complex real ext-real Element of REAL
1 + |.((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>))).| is complex real ext-real Element of REAL
(Partial_Sums (0c)) . 0 is complex Element of COMPLEX
(Partial_Sums (0c)) . 0 is complex set
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (0c)) . bq is complex set
bq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (0c)) . (bq + 1) is complex set
(Partial_Sums (0c)) . bq is complex Element of COMPLEX
(Partial_Sums (0c)) . (bq + 1) is complex Element of COMPLEX
(0c) . (bq + 1) is complex Element of COMPLEX
1r + ((0c) . (bq + 1)) is complex Element of COMPLEX
(0c) . bq is complex Element of COMPLEX
((0c) . bq) * 0c is complex Element of COMPLEX
(bq + 1) + (0 * <i>) is complex set
(((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>)) is complex Element of COMPLEX
((bq + 1) + (0 * <i>)) " is complex set
(((0c) . bq) * 0c) * (((bq + 1) + (0 * <i>)) ") is complex set
1r + ((((0c) . bq) * 0c) / ((bq + 1) + (0 * <i>))) is complex Element of COMPLEX
bq is complex set
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq) . 0 is complex set
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq) . bq is complex set
bq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq) . (bq + 1) is complex set
(bq) . bq is complex Element of COMPLEX
(bq) . (bq + 1) is complex Element of COMPLEX
((bq) . bq) * bq is complex set
(bq + 1) + (0 * <i>) is complex set
(((bq) . bq) * bq) / ((bq + 1) + (0 * <i>)) is complex Element of COMPLEX
((bq + 1) + (0 * <i>)) " is complex set
(((bq) . bq) * bq) * (((bq + 1) + (0 * <i>)) ") is complex set
0c / bq is complex Element of COMPLEX
bq " is complex set
0c * (bq ") is complex set
(((bq) . bq) * bq) / bq is complex Element of COMPLEX
(((bq) . bq) * bq) * (bq ") is complex set
bq / bq is complex Element of COMPLEX
bq * (bq ") is complex set
((bq) . bq) * (bq / bq) is complex Element of COMPLEX
((bq) . bq) * 1 is complex set
|.(bq).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
bq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
bq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . a is complex real ext-real Element of REAL
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
|.(bq).| . (a + 1) is complex real ext-real Element of REAL
|.(bq).| . a is complex real ext-real Element of REAL
(|.(bq).| . (a + 1)) / (|.(bq).| . a) is complex real ext-real Element of REAL
(|.(bq).| . a) " is complex real ext-real set
(|.(bq).| . (a + 1)) * ((|.(bq).| . a) ") is complex real ext-real set
(bq) . (a + 1) is complex Element of COMPLEX
|.((bq) . (a + 1)).| is complex real ext-real Element of REAL
Re ((bq) . (a + 1)) is complex real ext-real Element of REAL
(Re ((bq) . (a + 1))) ^2 is complex real ext-real Element of REAL
(Re ((bq) . (a + 1))) * (Re ((bq) . (a + 1))) is complex real ext-real set
Im ((bq) . (a + 1)) is complex real ext-real Element of REAL
(Im ((bq) . (a + 1))) ^2 is complex real ext-real Element of REAL
(Im ((bq) . (a + 1))) * (Im ((bq) . (a + 1))) is complex real ext-real set
((Re ((bq) . (a + 1))) ^2) + ((Im ((bq) . (a + 1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . (a + 1))) ^2) + ((Im ((bq) . (a + 1))) ^2)) is complex real ext-real Element of REAL
|.((bq) . (a + 1)).| / (|.(bq).| . a) is complex real ext-real Element of REAL
|.((bq) . (a + 1)).| * ((|.(bq).| . a) ") is complex real ext-real set
(bq) . a is complex Element of COMPLEX
|.((bq) . a).| is complex real ext-real Element of REAL
Re ((bq) . a) is complex real ext-real Element of REAL
(Re ((bq) . a)) ^2 is complex real ext-real Element of REAL
(Re ((bq) . a)) * (Re ((bq) . a)) is complex real ext-real set
Im ((bq) . a) is complex real ext-real Element of REAL
(Im ((bq) . a)) ^2 is complex real ext-real Element of REAL
(Im ((bq) . a)) * (Im ((bq) . a)) is complex real ext-real set
((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2)) is complex real ext-real Element of REAL
|.((bq) . (a + 1)).| / |.((bq) . a).| is complex real ext-real Element of REAL
|.((bq) . a).| " is complex real ext-real set
|.((bq) . (a + 1)).| * (|.((bq) . a).| ") is complex real ext-real set
((bq) . (a + 1)) / ((bq) . a) is complex Element of COMPLEX
((bq) . a) " is complex set
(Re ((bq) . a)) / (((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2)) is complex real ext-real Element of REAL
- (Im ((bq) . a)) is complex real ext-real Element of REAL
(- (Im ((bq) . a))) / (((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2)) is complex real ext-real Element of REAL
((- (Im ((bq) . a))) / (((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2))) * <i> is complex set
((Re ((bq) . a)) / (((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2))) + (((- (Im ((bq) . a))) / (((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2))) * <i>) is complex set
((bq) . (a + 1)) * (((bq) . a) ") is complex set
|.(((bq) . (a + 1)) / ((bq) . a)).| is complex real ext-real Element of REAL
Re (((bq) . (a + 1)) / ((bq) . a)) is complex real ext-real Element of REAL
(Re (((bq) . (a + 1)) / ((bq) . a))) ^2 is complex real ext-real Element of REAL
(Re (((bq) . (a + 1)) / ((bq) . a))) * (Re (((bq) . (a + 1)) / ((bq) . a))) is complex real ext-real set
Im (((bq) . (a + 1)) / ((bq) . a)) is complex real ext-real Element of REAL
(Im (((bq) . (a + 1)) / ((bq) . a))) ^2 is complex real ext-real Element of REAL
(Im (((bq) . (a + 1)) / ((bq) . a))) * (Im (((bq) . (a + 1)) / ((bq) . a))) is complex real ext-real set
((Re (((bq) . (a + 1)) / ((bq) . a))) ^2) + ((Im (((bq) . (a + 1)) / ((bq) . a))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((bq) . (a + 1)) / ((bq) . a))) ^2) + ((Im (((bq) . (a + 1)) / ((bq) . a))) ^2)) is complex real ext-real Element of REAL
((bq) . a) * bq is complex set
(a + 1) + (0 * <i>) is complex set
(((bq) . a) * bq) / ((a + 1) + (0 * <i>)) is complex Element of COMPLEX
((a + 1) + (0 * <i>)) " is complex set
(((bq) . a) * bq) * (((a + 1) + (0 * <i>)) ") is complex set
((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a) is complex Element of COMPLEX
((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) * (((bq) . a) ") is complex set
|.(((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a)).| is complex real ext-real Element of REAL
Re (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a)) is complex real ext-real Element of REAL
(Re (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a))) ^2 is complex real ext-real Element of REAL
(Re (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a))) * (Re (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a))) is complex real ext-real set
Im (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a)) is complex real ext-real Element of REAL
(Im (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a))) ^2 is complex real ext-real Element of REAL
(Im (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a))) * (Im (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a))) is complex real ext-real set
((Re (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a))) ^2) + ((Im (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a))) ^2) + ((Im (((((bq) . a) * bq) / ((a + 1) + (0 * <i>))) / ((bq) . a))) ^2)) is complex real ext-real Element of REAL
bq / ((a + 1) + (0 * <i>)) is complex Element of COMPLEX
bq * (((a + 1) + (0 * <i>)) ") is complex set
((bq) . a) * (bq / ((a + 1) + (0 * <i>))) is complex Element of COMPLEX
(((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a) is complex Element of COMPLEX
(((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) * (((bq) . a) ") is complex set
|.((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a)).| is complex real ext-real Element of REAL
Re ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a)) is complex real ext-real Element of REAL
(Re ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a))) ^2 is complex real ext-real Element of REAL
(Re ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a))) * (Re ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a))) is complex real ext-real set
Im ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a)) is complex real ext-real Element of REAL
(Im ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a))) ^2 is complex real ext-real Element of REAL
(Im ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a))) * (Im ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a))) is complex real ext-real set
((Re ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a))) ^2) + ((Im ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a))) ^2) + ((Im ((((bq) . a) * (bq / ((a + 1) + (0 * <i>)))) / ((bq) . a))) ^2)) is complex real ext-real Element of REAL
|.(bq / ((a + 1) + (0 * <i>))).| is complex real ext-real Element of REAL
Re (bq / ((a + 1) + (0 * <i>))) is complex real ext-real Element of REAL
(Re (bq / ((a + 1) + (0 * <i>)))) ^2 is complex real ext-real Element of REAL
(Re (bq / ((a + 1) + (0 * <i>)))) * (Re (bq / ((a + 1) + (0 * <i>)))) is complex real ext-real set
Im (bq / ((a + 1) + (0 * <i>))) is complex real ext-real Element of REAL
(Im (bq / ((a + 1) + (0 * <i>)))) ^2 is complex real ext-real Element of REAL
(Im (bq / ((a + 1) + (0 * <i>)))) * (Im (bq / ((a + 1) + (0 * <i>)))) is complex real ext-real set
((Re (bq / ((a + 1) + (0 * <i>)))) ^2) + ((Im (bq / ((a + 1) + (0 * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (bq / ((a + 1) + (0 * <i>)))) ^2) + ((Im (bq / ((a + 1) + (0 * <i>)))) ^2)) is complex real ext-real Element of REAL
|.bq.| is complex real ext-real Element of REAL
Re bq is complex real ext-real Element of REAL
(Re bq) ^2 is complex real ext-real Element of REAL
(Re bq) * (Re bq) is complex real ext-real set
Im bq is complex real ext-real Element of REAL
(Im bq) ^2 is complex real ext-real Element of REAL
(Im bq) * (Im bq) is complex real ext-real set
((Re bq) ^2) + ((Im bq) ^2) is complex real ext-real Element of REAL
sqrt (((Re bq) ^2) + ((Im bq) ^2)) is complex real ext-real Element of REAL
abs (a + 1) is complex real ext-real V63() Element of REAL
Re (a + 1) is complex real ext-real Element of REAL
(Re (a + 1)) ^2 is complex real ext-real Element of REAL
(Re (a + 1)) * (Re (a + 1)) is complex real ext-real set
Im (a + 1) is complex real ext-real Element of REAL
(Im (a + 1)) ^2 is complex real ext-real Element of REAL
(Im (a + 1)) * (Im (a + 1)) is complex real ext-real set
((Re (a + 1)) ^2) + ((Im (a + 1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (a + 1)) ^2) + ((Im (a + 1)) ^2)) is complex real ext-real Element of REAL
|.bq.| / (abs (a + 1)) is complex real ext-real Element of REAL
(abs (a + 1)) " is complex real ext-real set
|.bq.| * ((abs (a + 1)) ") is complex real ext-real set
|.bq.| / (a + 1) is complex real ext-real Element of REAL
(a + 1) " is non empty complex real ext-real positive non negative set
|.bq.| * ((a + 1) ") is complex real ext-real set
lim bq is complex real ext-real Element of REAL
bq is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
(bq) . 0 is complex Element of COMPLEX
bq is complex Element of COMPLEX
(0,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(0,bq,bq) . 0 is complex Element of COMPLEX
bq |^ 0 is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . 0 is complex Element of COMPLEX
(bq |^ 0) / (0 !) is complex Element of COMPLEX
(0 !) " is non empty complex real ext-real positive non negative set
(bq |^ 0) * ((0 !) ") is complex set
0 -' 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(0) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(0) . 0 is complex Element of COMPLEX
((0) . 0) * (bq |^ 0) is complex set
bq |^ 0 is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . 0 is complex Element of COMPLEX
(((0) . 0) * (bq |^ 0)) * (bq |^ 0) is complex set
1 * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
1 / (1 * 1) is non empty complex real ext-real positive non negative Element of REAL
(1 * 1) " is non empty complex real ext-real positive non negative set
1 * ((1 * 1) ") is non empty complex real ext-real positive non negative set
(1 / (1 * 1)) * (bq |^ 0) is complex set
((1 / (1 * 1)) * (bq |^ 0)) * (bq |^ 0) is complex set
1r * ((bq GeoSeq) . 0) is complex Element of COMPLEX
bq is complex Element of COMPLEX
bq is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((b + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((b + 1),bq,bq) . a is complex Element of COMPLEX
(b,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(b,bq,bq) . a is complex Element of COMPLEX
((b + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((b + 1),bq,bq) . a is complex Element of COMPLEX
((b,bq,bq) . a) + (((b + 1),bq,bq) . a) is complex Element of COMPLEX
(b + 1) -' a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(b + 1) - a is complex real ext-real Element of REAL
- a is complex real ext-real non positive set
(b + 1) + (- a) is complex real ext-real set
b - a is complex real ext-real Element of REAL
b + (- a) is complex real ext-real set
(b - a) + 1 is complex real ext-real Element of REAL
b -' a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(b -' a) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
(bq) . a is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (bq)) . ((b -' a) + 1) is complex Element of COMPLEX
((bq) . a) * ((Partial_Sums (bq)) . ((b -' a) + 1)) is complex Element of COMPLEX
(Partial_Sums (bq)) . (b -' a) is complex Element of COMPLEX
(bq) . ((b + 1) -' a) is complex Element of COMPLEX
((Partial_Sums (bq)) . (b -' a)) + ((bq) . ((b + 1) -' a)) is complex Element of COMPLEX
((bq) . a) * (((Partial_Sums (bq)) . (b -' a)) + ((bq) . ((b + 1) -' a))) is complex Element of COMPLEX
((bq) . a) * ((Partial_Sums (bq)) . (b -' a)) is complex Element of COMPLEX
((bq) . a) * ((bq) . ((b + 1) -' a)) is complex Element of COMPLEX
(((bq) . a) * ((Partial_Sums (bq)) . (b -' a))) + (((bq) . a) * ((bq) . ((b + 1) -' a))) is complex Element of COMPLEX
((b,bq,bq) . a) + (((bq) . a) * ((bq) . ((b + 1) -' a))) is complex Element of COMPLEX
bq |^ a is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . a is complex Element of COMPLEX
a ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . a is complex real ext-real Element of REAL
(bq |^ a) / (a !) is complex Element of COMPLEX
(a !) " is non empty complex real ext-real positive non negative set
(bq |^ a) * ((a !) ") is complex set
((bq |^ a) / (a !)) * ((bq) . ((b + 1) -' a)) is complex Element of COMPLEX
bq |^ ((b + 1) -' a) is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . ((b + 1) -' a) is complex Element of COMPLEX
((b + 1) -' a) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((b + 1) -' a) is complex real ext-real Element of REAL
(bq |^ ((b + 1) -' a)) / (((b + 1) -' a) !) is complex Element of COMPLEX
(((b + 1) -' a) !) " is non empty complex real ext-real positive non negative set
(bq |^ ((b + 1) -' a)) * ((((b + 1) -' a) !) ") is complex set
((bq |^ a) / (a !)) * ((bq |^ ((b + 1) -' a)) / (((b + 1) -' a) !)) is complex Element of COMPLEX
(bq |^ a) * (bq |^ ((b + 1) -' a)) is complex set
((bq |^ a) * (bq |^ ((b + 1) -' a))) * 1r is complex set
(a !) * (((b + 1) -' a) !) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
(((bq |^ a) * (bq |^ ((b + 1) -' a))) * 1r) / ((a !) * (((b + 1) -' a) !)) is complex Element of COMPLEX
((a !) * (((b + 1) -' a) !)) " is non empty complex real ext-real positive non negative set
(((bq |^ a) * (bq |^ ((b + 1) -' a))) * 1r) * (((a !) * (((b + 1) -' a) !)) ") is complex set
1r / ((a !) * (((b + 1) -' a) !)) is complex Element of COMPLEX
1r * (((a !) * (((b + 1) -' a) !)) ") is complex set
((bq |^ a) * (bq |^ ((b + 1) -' a))) * (1r / ((a !) * (((b + 1) -' a) !))) is complex set
((b + 1)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((b + 1)) . a is complex Element of COMPLEX
(((b + 1)) . a) * ((bq |^ a) * (bq |^ ((b + 1) -' a))) is complex set
(((b + 1)) . a) * (bq |^ a) is complex set
((((b + 1)) . a) * (bq |^ a)) * (bq |^ ((b + 1) -' a)) is complex set
bq is complex Element of COMPLEX
bq is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((a + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((a + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((a + 1),bq,bq)) . a is complex Element of COMPLEX
(a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (a,bq,bq)) . a is complex Element of COMPLEX
((a + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((a + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((a + 1),bq,bq)) . a is complex Element of COMPLEX
((Partial_Sums (a,bq,bq)) . a) + ((Partial_Sums ((a + 1),bq,bq)) . a) is complex Element of COMPLEX
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((a + 1),bq,bq) . b is complex Element of COMPLEX
(a,bq,bq) . b is complex Element of COMPLEX
((a + 1),bq,bq) . b is complex Element of COMPLEX
((a,bq,bq) . b) + (((a + 1),bq,bq) . b) is complex Element of COMPLEX
(a,bq,bq) + ((a + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((a,bq,bq) + ((a + 1),bq,bq)) . b is complex Element of COMPLEX
Partial_Sums ((a,bq,bq) + ((a + 1),bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((a,bq,bq) + ((a + 1),bq,bq))) . a is complex Element of COMPLEX
(Partial_Sums (a,bq,bq)) + (Partial_Sums ((a + 1),bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((Partial_Sums (a,bq,bq)) + (Partial_Sums ((a + 1),bq,bq))) . a is complex Element of COMPLEX
bq is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
bq is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq) . a is complex Element of COMPLEX
(a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a,bq,bq) . a is complex Element of COMPLEX
a - a is complex real ext-real Element of REAL
- a is complex real ext-real non positive set
a + (- a) is complex real ext-real set
a -' a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a -' a) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (a -' a) is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a) . a is complex Element of COMPLEX
bq |^ a is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . a is complex Element of COMPLEX
((a) . a) * (bq |^ a) is complex set
bq |^ 0 is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . 0 is complex Element of COMPLEX
(((a) . a) * (bq |^ a)) * (bq |^ 0) is complex set
(((a) . a) * (bq |^ a)) * 1r is complex set
a ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . a is complex real ext-real Element of REAL
(a !) * 1r is complex set
1r / ((a !) * 1r) is complex Element of COMPLEX
((a !) * 1r) " is complex set
1r * (((a !) * 1r) ") is complex set
(1r / ((a !) * 1r)) * (bq |^ a) is complex set
(bq |^ a) * 1r is complex set
((bq |^ a) * 1r) / (a !) is complex Element of COMPLEX
(a !) " is non empty complex real ext-real positive non negative set
((bq |^ a) * 1r) * ((a !) ") is complex set
bq is complex Element of COMPLEX
bq is complex Element of COMPLEX
bq + bq is complex Element of COMPLEX
((bq + bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((bq + bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums ((bq + bq))) . a is complex Element of COMPLEX
(a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (a,bq,bq)) . a is complex Element of COMPLEX
(Partial_Sums ((bq + bq))) . 0 is complex Element of COMPLEX
((bq + bq)) . 0 is complex Element of COMPLEX
0 -' 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(0,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (0,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (0,bq,bq)) . 0 is complex Element of COMPLEX
(0,bq,bq) . 0 is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
(bq) . 0 is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (bq)) . 0 is complex Element of COMPLEX
((bq) . 0) * ((Partial_Sums (bq)) . 0) is complex Element of COMPLEX
(bq) . 0 is complex Element of COMPLEX
((bq) . 0) * ((bq) . 0) is complex Element of COMPLEX
1r * ((bq) . 0) is complex Element of COMPLEX
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums ((bq + bq))) . b is complex Element of COMPLEX
(b,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (b,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (b,bq,bq)) . b is complex Element of COMPLEX
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums ((bq + bq))) . (b + 1) is complex Element of COMPLEX
((b + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((b + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((b + 1),bq,bq)) . (b + 1) is complex Element of COMPLEX
(Partial_Sums ((b + 1),bq,bq)) . b is complex Element of COMPLEX
((b + 1),bq,bq) . (b + 1) is complex Element of COMPLEX
((Partial_Sums ((b + 1),bq,bq)) . b) + (((b + 1),bq,bq) . (b + 1)) is complex Element of COMPLEX
((b + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((b + 1),bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((b + 1),bq,bq)) . b is complex Element of COMPLEX
((Partial_Sums (b,bq,bq)) . b) + ((Partial_Sums ((b + 1),bq,bq)) . b) is complex Element of COMPLEX
(((Partial_Sums (b,bq,bq)) . b) + ((Partial_Sums ((b + 1),bq,bq)) . b)) + (((b + 1),bq,bq) . (b + 1)) is complex Element of COMPLEX
((Partial_Sums ((b + 1),bq,bq)) . b) + (((b + 1),bq,bq) . (b + 1)) is complex Element of COMPLEX
((Partial_Sums ((bq + bq))) . b) + (((Partial_Sums ((b + 1),bq,bq)) . b) + (((b + 1),bq,bq) . (b + 1))) is complex Element of COMPLEX
(b + 1) -' (b + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq) . (b + 1) is complex Element of COMPLEX
((bq) . (b + 1)) * ((Partial_Sums (bq)) . 0) is complex Element of COMPLEX
((bq) . (b + 1)) * ((bq) . 0) is complex Element of COMPLEX
((bq) . (b + 1)) * 1 is complex set
((b + 1),bq,bq) . (b + 1) is complex Element of COMPLEX
(Partial_Sums ((b + 1),bq,bq)) . (b + 1) is complex Element of COMPLEX
(bq + bq) |^ (b + 1) is complex set
(bq + bq) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq + bq) GeoSeq) . (b + 1) is complex Element of COMPLEX
(b + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (b + 1) is complex real ext-real Element of REAL
((bq + bq) |^ (b + 1)) / ((b + 1) !) is complex Element of COMPLEX
((b + 1) !) " is non empty complex real ext-real positive non negative set
((bq + bq) |^ (b + 1)) * (((b + 1) !) ") is complex set
((bq + bq)) . (b + 1) is complex Element of COMPLEX
((Partial_Sums ((bq + bq))) . b) + (((bq + bq)) . (b + 1)) is complex Element of COMPLEX
bq is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
bq is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
bq + bq is complex Element of COMPLEX
((bq + bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((bq + bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . a is complex Element of COMPLEX
(Partial_Sums (bq)) . a is complex Element of COMPLEX
((Partial_Sums (bq)) . a) * ((Partial_Sums (bq)) . a) is complex Element of COMPLEX
(Partial_Sums ((bq + bq))) . a is complex Element of COMPLEX
(((Partial_Sums (bq)) . a) * ((Partial_Sums (bq)) . a)) - ((Partial_Sums ((bq + bq))) . a) is complex Element of COMPLEX
- ((Partial_Sums ((bq + bq))) . a) is complex set
(((Partial_Sums (bq)) . a) * ((Partial_Sums (bq)) . a)) + (- ((Partial_Sums ((bq + bq))) . a)) is complex set
(bq,bq,a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq,bq,a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (bq,bq,a)) . a is complex Element of COMPLEX
(a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (a,bq,bq)) . a is complex Element of COMPLEX
(((Partial_Sums (bq)) . a) * ((Partial_Sums (bq)) . a)) - ((Partial_Sums (a,bq,bq)) . a) is complex Element of COMPLEX
- ((Partial_Sums (a,bq,bq)) . a) is complex set
(((Partial_Sums (bq)) . a) * ((Partial_Sums (bq)) . a)) + (- ((Partial_Sums (a,bq,bq)) . a)) is complex set
((Partial_Sums (bq)) . a) (#) (Partial_Sums (bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
(((Partial_Sums (bq)) . a) (#) (Partial_Sums (bq))) . a is complex Element of COMPLEX
((((Partial_Sums (bq)) . a) (#) (Partial_Sums (bq))) . a) - ((Partial_Sums (a,bq,bq)) . a) is complex Element of COMPLEX
((((Partial_Sums (bq)) . a) (#) (Partial_Sums (bq))) . a) + (- ((Partial_Sums (a,bq,bq)) . a)) is complex set
((Partial_Sums (bq)) . a) (#) (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums (((Partial_Sums (bq)) . a) (#) (bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (((Partial_Sums (bq)) . a) (#) (bq))) . a is complex Element of COMPLEX
((Partial_Sums (((Partial_Sums (bq)) . a) (#) (bq))) . a) - ((Partial_Sums (a,bq,bq)) . a) is complex Element of COMPLEX
((Partial_Sums (((Partial_Sums (bq)) . a) (#) (bq))) . a) + (- ((Partial_Sums (a,bq,bq)) . a)) is complex set
- (Partial_Sums (a,bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(- 1) (#) (Partial_Sums (a,bq,bq)) is Relation-like NAT -defined Function-like total complex-valued set
(- (Partial_Sums (a,bq,bq))) . a is complex Element of COMPLEX
((Partial_Sums (((Partial_Sums (bq)) . a) (#) (bq))) . a) + ((- (Partial_Sums (a,bq,bq))) . a) is complex Element of COMPLEX
(Partial_Sums (((Partial_Sums (bq)) . a) (#) (bq))) - (Partial_Sums (a,bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
- (Partial_Sums (a,bq,bq)) is Relation-like NAT -defined Function-like total complex-valued set
(Partial_Sums (((Partial_Sums (bq)) . a) (#) (bq))) + (- (Partial_Sums (a,bq,bq))) is Relation-like NAT -defined Function-like total complex-valued set
((Partial_Sums (((Partial_Sums (bq)) . a) (#) (bq))) - (Partial_Sums (a,bq,bq))) . a is complex Element of COMPLEX
(((Partial_Sums (bq)) . a) (#) (bq)) - (a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
- (a,bq,bq) is Relation-like NAT -defined Function-like total complex-valued set
(- 1) (#) (a,bq,bq) is Relation-like NAT -defined Function-like total complex-valued set
(((Partial_Sums (bq)) . a) (#) (bq)) + (- (a,bq,bq)) is Relation-like NAT -defined Function-like total complex-valued set
Partial_Sums ((((Partial_Sums (bq)) . a) (#) (bq)) - (a,bq,bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((((Partial_Sums (bq)) . a) (#) (bq)) - (a,bq,bq))) . a is complex Element of COMPLEX
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((((Partial_Sums (bq)) . a) (#) (bq)) - (a,bq,bq)) . b is complex Element of COMPLEX
(bq,bq,a) . b is complex Element of COMPLEX
(((Partial_Sums (bq)) . a) (#) (bq)) . b is complex Element of COMPLEX
- (a,bq,bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(- (a,bq,bq)) . b is complex Element of COMPLEX
((((Partial_Sums (bq)) . a) (#) (bq)) . b) + ((- (a,bq,bq)) . b) is complex Element of COMPLEX
(a,bq,bq) . b is complex Element of COMPLEX
((((Partial_Sums (bq)) . a) (#) (bq)) . b) - ((a,bq,bq) . b) is complex Element of COMPLEX
- ((a,bq,bq) . b) is complex set
((((Partial_Sums (bq)) . a) (#) (bq)) . b) + (- ((a,bq,bq) . b)) is complex set
(bq) . b is complex Element of COMPLEX
((Partial_Sums (bq)) . a) * ((bq) . b) is complex Element of COMPLEX
(((Partial_Sums (bq)) . a) * ((bq) . b)) - ((a,bq,bq) . b) is complex Element of COMPLEX
(((Partial_Sums (bq)) . a) * ((bq) . b)) + (- ((a,bq,bq) . b)) is complex set
((bq) . b) * ((Partial_Sums (bq)) . a) is complex Element of COMPLEX
a -' b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . (a -' b) is complex Element of COMPLEX
((bq) . b) * ((Partial_Sums (bq)) . (a -' b)) is complex Element of COMPLEX
(((bq) . b) * ((Partial_Sums (bq)) . a)) - (((bq) . b) * ((Partial_Sums (bq)) . (a -' b))) is complex Element of COMPLEX
- (((bq) . b) * ((Partial_Sums (bq)) . (a -' b))) is complex set
(((bq) . b) * ((Partial_Sums (bq)) . a)) + (- (((bq) . b) * ((Partial_Sums (bq)) . (a -' b)))) is complex set
((Partial_Sums (bq)) . a) - ((Partial_Sums (bq)) . (a -' b)) is complex Element of COMPLEX
- ((Partial_Sums (bq)) . (a -' b)) is complex set
((Partial_Sums (bq)) . a) + (- ((Partial_Sums (bq)) . (a -' b))) is complex set
((bq) . b) * (((Partial_Sums (bq)) . a) - ((Partial_Sums (bq)) . (a -' b))) is complex Element of COMPLEX
bq is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
|.bq.| is complex real ext-real Element of REAL
Re bq is complex real ext-real Element of REAL
(Re bq) ^2 is complex real ext-real Element of REAL
(Re bq) * (Re bq) is complex real ext-real set
Im bq is complex real ext-real Element of REAL
(Im bq) ^2 is complex real ext-real Element of REAL
(Im bq) * (Im bq) is complex real ext-real set
((Re bq) ^2) + ((Im bq) ^2) is complex real ext-real Element of REAL
sqrt (((Re bq) ^2) + ((Im bq) ^2)) is complex real ext-real Element of REAL
(|.bq.|) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (|.bq.|) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (|.bq.|) is complex real ext-real Element of REAL
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . bq is complex Element of COMPLEX
|.((Partial_Sums (bq)) . bq).| is complex real ext-real Element of REAL
Re ((Partial_Sums (bq)) . bq) is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . bq)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . bq)) * (Re ((Partial_Sums (bq)) . bq)) is complex real ext-real set
Im ((Partial_Sums (bq)) . bq) is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . bq)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . bq)) * (Im ((Partial_Sums (bq)) . bq)) is complex real ext-real set
((Re ((Partial_Sums (bq)) . bq)) ^2) + ((Im ((Partial_Sums (bq)) . bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums (bq)) . bq)) ^2) + ((Im ((Partial_Sums (bq)) . bq)) ^2)) is complex real ext-real Element of REAL
(Partial_Sums (|.bq.|)) . bq is complex real ext-real Element of REAL
(Partial_Sums (bq)) . 0 is complex Element of COMPLEX
|.((Partial_Sums (bq)) . 0).| is complex real ext-real Element of REAL
Re ((Partial_Sums (bq)) . 0) is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . 0)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . 0)) * (Re ((Partial_Sums (bq)) . 0)) is complex real ext-real set
Im ((Partial_Sums (bq)) . 0) is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . 0)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . 0)) * (Im ((Partial_Sums (bq)) . 0)) is complex real ext-real set
((Re ((Partial_Sums (bq)) . 0)) ^2) + ((Im ((Partial_Sums (bq)) . 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums (bq)) . 0)) ^2) + ((Im ((Partial_Sums (bq)) . 0)) ^2)) is complex real ext-real Element of REAL
(bq) . 0 is complex Element of COMPLEX
|.((bq) . 0).| is complex real ext-real Element of REAL
Re ((bq) . 0) is complex real ext-real Element of REAL
(Re ((bq) . 0)) ^2 is complex real ext-real Element of REAL
(Re ((bq) . 0)) * (Re ((bq) . 0)) is complex real ext-real set
Im ((bq) . 0) is complex real ext-real Element of REAL
(Im ((bq) . 0)) ^2 is complex real ext-real Element of REAL
(Im ((bq) . 0)) * (Im ((bq) . 0)) is complex real ext-real set
((Re ((bq) . 0)) ^2) + ((Im ((bq) . 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . 0)) ^2) + ((Im ((bq) . 0)) ^2)) is complex real ext-real Element of REAL
bq |^ 0 is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . 0 is complex Element of COMPLEX
(bq |^ 0) / (0 !) is complex Element of COMPLEX
(0 !) " is non empty complex real ext-real positive non negative set
(bq |^ 0) * ((0 !) ") is complex set
|.((bq |^ 0) / (0 !)).| is complex real ext-real Element of REAL
Re ((bq |^ 0) / (0 !)) is complex real ext-real Element of REAL
(Re ((bq |^ 0) / (0 !))) ^2 is complex real ext-real Element of REAL
(Re ((bq |^ 0) / (0 !))) * (Re ((bq |^ 0) / (0 !))) is complex real ext-real set
Im ((bq |^ 0) / (0 !)) is complex real ext-real Element of REAL
(Im ((bq |^ 0) / (0 !))) ^2 is complex real ext-real Element of REAL
(Im ((bq |^ 0) / (0 !))) * (Im ((bq |^ 0) / (0 !))) is complex real ext-real set
((Re ((bq |^ 0) / (0 !))) ^2) + ((Im ((bq |^ 0) / (0 !))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq |^ 0) / (0 !))) ^2) + ((Im ((bq |^ 0) / (0 !))) ^2)) is complex real ext-real Element of REAL
(Partial_Sums (|.bq.|)) . 0 is complex real ext-real Element of REAL
(|.bq.|) . 0 is complex real ext-real Element of REAL
|.bq.| |^ 0 is complex real ext-real Element of REAL
|.bq.| GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(|.bq.| GeoSeq) . 0 is complex Element of COMPLEX
(|.bq.| |^ 0) / (0 !) is complex real ext-real Element of REAL
(|.bq.| |^ 0) * ((0 !) ") is complex real ext-real set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . a is complex Element of COMPLEX
|.((Partial_Sums (bq)) . a).| is complex real ext-real Element of REAL
Re ((Partial_Sums (bq)) . a) is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . a)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . a)) * (Re ((Partial_Sums (bq)) . a)) is complex real ext-real set
Im ((Partial_Sums (bq)) . a) is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . a)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . a)) * (Im ((Partial_Sums (bq)) . a)) is complex real ext-real set
((Re ((Partial_Sums (bq)) . a)) ^2) + ((Im ((Partial_Sums (bq)) . a)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums (bq)) . a)) ^2) + ((Im ((Partial_Sums (bq)) . a)) ^2)) is complex real ext-real Element of REAL
(Partial_Sums (|.bq.|)) . a is complex real ext-real Element of REAL
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . (a + 1) is complex Element of COMPLEX
|.((Partial_Sums (bq)) . (a + 1)).| is complex real ext-real Element of REAL
Re ((Partial_Sums (bq)) . (a + 1)) is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . (a + 1))) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . (a + 1))) * (Re ((Partial_Sums (bq)) . (a + 1))) is complex real ext-real set
Im ((Partial_Sums (bq)) . (a + 1)) is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . (a + 1))) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . (a + 1))) * (Im ((Partial_Sums (bq)) . (a + 1))) is complex real ext-real set
((Re ((Partial_Sums (bq)) . (a + 1))) ^2) + ((Im ((Partial_Sums (bq)) . (a + 1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums (bq)) . (a + 1))) ^2) + ((Im ((Partial_Sums (bq)) . (a + 1))) ^2)) is complex real ext-real Element of REAL
(Partial_Sums (|.bq.|)) . (a + 1) is complex real ext-real Element of REAL
(bq) . (a + 1) is complex Element of COMPLEX
((Partial_Sums (bq)) . a) + ((bq) . (a + 1)) is complex Element of COMPLEX
|.(((Partial_Sums (bq)) . a) + ((bq) . (a + 1))).| is complex real ext-real Element of REAL
Re (((Partial_Sums (bq)) . a) + ((bq) . (a + 1))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . a) + ((bq) . (a + 1)))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . a) + ((bq) . (a + 1)))) * (Re (((Partial_Sums (bq)) . a) + ((bq) . (a + 1)))) is complex real ext-real set
Im (((Partial_Sums (bq)) . a) + ((bq) . (a + 1))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . a) + ((bq) . (a + 1)))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . a) + ((bq) . (a + 1)))) * (Im (((Partial_Sums (bq)) . a) + ((bq) . (a + 1)))) is complex real ext-real set
((Re (((Partial_Sums (bq)) . a) + ((bq) . (a + 1)))) ^2) + ((Im (((Partial_Sums (bq)) . a) + ((bq) . (a + 1)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (bq)) . a) + ((bq) . (a + 1)))) ^2) + ((Im (((Partial_Sums (bq)) . a) + ((bq) . (a + 1)))) ^2)) is complex real ext-real Element of REAL
|.((bq) . (a + 1)).| is complex real ext-real Element of REAL
Re ((bq) . (a + 1)) is complex real ext-real Element of REAL
(Re ((bq) . (a + 1))) ^2 is complex real ext-real Element of REAL
(Re ((bq) . (a + 1))) * (Re ((bq) . (a + 1))) is complex real ext-real set
Im ((bq) . (a + 1)) is complex real ext-real Element of REAL
(Im ((bq) . (a + 1))) ^2 is complex real ext-real Element of REAL
(Im ((bq) . (a + 1))) * (Im ((bq) . (a + 1))) is complex real ext-real set
((Re ((bq) . (a + 1))) ^2) + ((Im ((bq) . (a + 1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . (a + 1))) ^2) + ((Im ((bq) . (a + 1))) ^2)) is complex real ext-real Element of REAL
|.((Partial_Sums (bq)) . a).| + |.((bq) . (a + 1)).| is complex real ext-real Element of REAL
(|.bq.|) . (a + 1) is complex real ext-real Element of REAL
|.((Partial_Sums (bq)) . a).| + ((|.bq.|) . (a + 1)) is complex real ext-real Element of REAL
((Partial_Sums (|.bq.|)) . a) + ((|.bq.|) . (a + 1)) is complex real ext-real Element of REAL
a is set
|.(bq).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
|.(bq).| . a is complex real ext-real Element of REAL
(bq) . a is complex set
|.((bq) . a).| is complex real ext-real Element of REAL
Re ((bq) . a) is complex real ext-real Element of REAL
(Re ((bq) . a)) ^2 is complex real ext-real Element of REAL
(Re ((bq) . a)) * (Re ((bq) . a)) is complex real ext-real set
Im ((bq) . a) is complex real ext-real Element of REAL
(Im ((bq) . a)) ^2 is complex real ext-real Element of REAL
(Im ((bq) . a)) * (Im ((bq) . a)) is complex real ext-real set
((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . a)) ^2) + ((Im ((bq) . a)) ^2)) is complex real ext-real Element of REAL
(|.bq.|) . a is complex real ext-real Element of REAL
lim (Partial_Sums (|.bq.|)) is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (|.bq.|)) . a is complex real ext-real Element of REAL
bq is complex Element of COMPLEX
|.bq.| is complex real ext-real Element of REAL
Re bq is complex real ext-real Element of REAL
(Re bq) ^2 is complex real ext-real Element of REAL
(Re bq) * (Re bq) is complex real ext-real set
Im bq is complex real ext-real Element of REAL
(Im bq) ^2 is complex real ext-real Element of REAL
(Im bq) * (Im bq) is complex real ext-real set
((Re bq) ^2) + ((Im bq) ^2) is complex real ext-real Element of REAL
sqrt (((Re bq) ^2) + ((Im bq) ^2)) is complex real ext-real Element of REAL
(|.bq.|) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (|.bq.|) is complex real ext-real Element of REAL
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (bq)) . 0 is complex Element of COMPLEX
|.((Partial_Sums (bq)) . 0).| is complex real ext-real Element of REAL
Re ((Partial_Sums (bq)) . 0) is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . 0)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . 0)) * (Re ((Partial_Sums (bq)) . 0)) is complex real ext-real set
Im ((Partial_Sums (bq)) . 0) is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . 0)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . 0)) * (Im ((Partial_Sums (bq)) . 0)) is complex real ext-real set
((Re ((Partial_Sums (bq)) . 0)) ^2) + ((Im ((Partial_Sums (bq)) . 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums (bq)) . 0)) ^2) + ((Im ((Partial_Sums (bq)) . 0)) ^2)) is complex real ext-real Element of REAL
(bq) . 0 is complex Element of COMPLEX
|.((bq) . 0).| is complex real ext-real Element of REAL
Re ((bq) . 0) is complex real ext-real Element of REAL
(Re ((bq) . 0)) ^2 is complex real ext-real Element of REAL
(Re ((bq) . 0)) * (Re ((bq) . 0)) is complex real ext-real set
Im ((bq) . 0) is complex real ext-real Element of REAL
(Im ((bq) . 0)) ^2 is complex real ext-real Element of REAL
(Im ((bq) . 0)) * (Im ((bq) . 0)) is complex real ext-real set
((Re ((bq) . 0)) ^2) + ((Im ((bq) . 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . 0)) ^2) + ((Im ((bq) . 0)) ^2)) is complex real ext-real Element of REAL
bq is complex Element of COMPLEX
|.bq.| is complex real ext-real Element of REAL
Re bq is complex real ext-real Element of REAL
(Re bq) ^2 is complex real ext-real Element of REAL
(Re bq) * (Re bq) is complex real ext-real set
Im bq is complex real ext-real Element of REAL
(Im bq) ^2 is complex real ext-real Element of REAL
(Im bq) * (Im bq) is complex real ext-real set
((Re bq) ^2) + ((Im bq) ^2) is complex real ext-real Element of REAL
sqrt (((Re bq) ^2) + ((Im bq) ^2)) is complex real ext-real Element of REAL
(|.bq.|) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(|.bq.|) . bq is complex real ext-real Element of REAL
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
(bq) . bq is complex Element of COMPLEX
|.((bq) . bq).| is complex real ext-real Element of REAL
Re ((bq) . bq) is complex real ext-real Element of REAL
(Re ((bq) . bq)) ^2 is complex real ext-real Element of REAL
(Re ((bq) . bq)) * (Re ((bq) . bq)) is complex real ext-real set
Im ((bq) . bq) is complex real ext-real Element of REAL
(Im ((bq) . bq)) ^2 is complex real ext-real Element of REAL
(Im ((bq) . bq)) * (Im ((bq) . bq)) is complex real ext-real set
((Re ((bq) . bq)) ^2) + ((Im ((bq) . bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . bq)) ^2) + ((Im ((bq) . bq)) ^2)) is complex real ext-real Element of REAL
bq is complex Element of COMPLEX
|.bq.| is complex real ext-real Element of REAL
Re bq is complex real ext-real Element of REAL
(Re bq) ^2 is complex real ext-real Element of REAL
(Re bq) * (Re bq) is complex real ext-real set
Im bq is complex real ext-real Element of REAL
(Im bq) ^2 is complex real ext-real Element of REAL
(Im bq) * (Im bq) is complex real ext-real set
((Re bq) ^2) + ((Im bq) ^2) is complex real ext-real Element of REAL
sqrt (((Re bq) ^2) + ((Im bq) ^2)) is complex real ext-real Element of REAL
(|.bq.|) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (|.bq.|) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (|.bq.|)) . bq is complex real ext-real Element of REAL
abs ((Partial_Sums (|.bq.|)) . bq) is complex real ext-real Element of REAL
Re ((Partial_Sums (|.bq.|)) . bq) is complex real ext-real Element of REAL
(Re ((Partial_Sums (|.bq.|)) . bq)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums (|.bq.|)) . bq)) * (Re ((Partial_Sums (|.bq.|)) . bq)) is complex real ext-real set
Im ((Partial_Sums (|.bq.|)) . bq) is complex real ext-real Element of REAL
(Im ((Partial_Sums (|.bq.|)) . bq)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums (|.bq.|)) . bq)) * (Im ((Partial_Sums (|.bq.|)) . bq)) is complex real ext-real set
((Re ((Partial_Sums (|.bq.|)) . bq)) ^2) + ((Im ((Partial_Sums (|.bq.|)) . bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums (|.bq.|)) . bq)) ^2) + ((Im ((Partial_Sums (|.bq.|)) . bq)) ^2)) is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (|.bq.|)) . a is complex real ext-real Element of REAL
((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq) is complex real ext-real Element of REAL
- ((Partial_Sums (|.bq.|)) . bq) is complex real ext-real set
((Partial_Sums (|.bq.|)) . a) + (- ((Partial_Sums (|.bq.|)) . bq)) is complex real ext-real set
abs (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq)) is complex real ext-real Element of REAL
Re (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq)) is complex real ext-real Element of REAL
(Re (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq))) * (Re (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq))) is complex real ext-real set
Im (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq)) is complex real ext-real Element of REAL
(Im (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq))) * (Im (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq))) is complex real ext-real set
((Re (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq))) ^2) + ((Im (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq))) ^2) + ((Im (((Partial_Sums (|.bq.|)) . a) - ((Partial_Sums (|.bq.|)) . bq))) ^2)) is complex real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(|.bq.|) . b is complex real ext-real Element of REAL
bq is complex Element of COMPLEX
bq is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq,bq,a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
|.(bq,bq,a).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums |.(bq,bq,a).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued non-decreasing Element of K19(K20(NAT,REAL))
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums |.(bq,bq,a).|) . b is complex real ext-real Element of REAL
abs ((Partial_Sums |.(bq,bq,a).|) . b) is complex real ext-real Element of REAL
Re ((Partial_Sums |.(bq,bq,a).|) . b) is complex real ext-real Element of REAL
(Re ((Partial_Sums |.(bq,bq,a).|) . b)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums |.(bq,bq,a).|) . b)) * (Re ((Partial_Sums |.(bq,bq,a).|) . b)) is complex real ext-real set
Im ((Partial_Sums |.(bq,bq,a).|) . b) is complex real ext-real Element of REAL
(Im ((Partial_Sums |.(bq,bq,a).|) . b)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums |.(bq,bq,a).|) . b)) * (Im ((Partial_Sums |.(bq,bq,a).|) . b)) is complex real ext-real set
((Re ((Partial_Sums |.(bq,bq,a).|) . b)) ^2) + ((Im ((Partial_Sums |.(bq,bq,a).|) . b)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums |.(bq,bq,a).|) . b)) ^2) + ((Im ((Partial_Sums |.(bq,bq,a).|) . b)) ^2)) is complex real ext-real Element of REAL
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
|.(bq,bq,a).| . th1 is complex real ext-real Element of REAL
(bq,bq,a) . th1 is complex Element of COMPLEX
|.((bq,bq,a) . th1).| is complex real ext-real Element of REAL
Re ((bq,bq,a) . th1) is complex real ext-real Element of REAL
(Re ((bq,bq,a) . th1)) ^2 is complex real ext-real Element of REAL
(Re ((bq,bq,a) . th1)) * (Re ((bq,bq,a) . th1)) is complex real ext-real set
Im ((bq,bq,a) . th1) is complex real ext-real Element of REAL
(Im ((bq,bq,a) . th1)) ^2 is complex real ext-real Element of REAL
(Im ((bq,bq,a) . th1)) * (Im ((bq,bq,a) . th1)) is complex real ext-real set
((Re ((bq,bq,a) . th1)) ^2) + ((Im ((bq,bq,a) . th1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq,bq,a) . th1)) ^2) + ((Im ((bq,bq,a) . th1)) ^2)) is complex real ext-real Element of REAL
(Partial_Sums |.(bq,bq,a).|) . 0 is complex real ext-real Element of REAL
|.(bq,bq,a).| . 0 is complex real ext-real Element of REAL
bq is complex Element of COMPLEX
bq is complex Element of COMPLEX
a is complex real ext-real set
|.bq.| is complex real ext-real Element of REAL
Re bq is complex real ext-real Element of REAL
(Re bq) ^2 is complex real ext-real Element of REAL
(Re bq) * (Re bq) is complex real ext-real set
Im bq is complex real ext-real Element of REAL
(Im bq) ^2 is complex real ext-real Element of REAL
(Im bq) * (Im bq) is complex real ext-real set
((Re bq) ^2) + ((Im bq) ^2) is complex real ext-real Element of REAL
sqrt (((Re bq) ^2) + ((Im bq) ^2)) is complex real ext-real Element of REAL
(|.bq.|) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (|.bq.|) is complex real ext-real Element of REAL
|.bq.| is complex real ext-real Element of REAL
Re bq is complex real ext-real Element of REAL
(Re bq) ^2 is complex real ext-real Element of REAL
(Re bq) * (Re bq) is complex real ext-real set
Im bq is complex real ext-real Element of REAL
(Im bq) ^2 is complex real ext-real Element of REAL
(Im bq) * (Im bq) is complex real ext-real set
((Re bq) ^2) + ((Im bq) ^2) is complex real ext-real Element of REAL
sqrt (((Re bq) ^2) + ((Im bq) ^2)) is complex real ext-real Element of REAL
(|.bq.|) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (|.bq.|) is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
4 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
4 * (Sum (|.bq.|)) is complex real ext-real Element of REAL
b / (4 * (Sum (|.bq.|))) is complex real ext-real Element of REAL
(4 * (Sum (|.bq.|))) " is complex real ext-real set
b * ((4 * (Sum (|.bq.|))) ") is complex real ext-real set
4 * (Sum (|.bq.|)) is complex real ext-real Element of REAL
b / (4 * (Sum (|.bq.|))) is complex real ext-real Element of REAL
(4 * (Sum (|.bq.|))) " is complex real ext-real set
b * ((4 * (Sum (|.bq.|))) ") is complex real ext-real set
min ((b / (4 * (Sum (|.bq.|)))),(b / (4 * (Sum (|.bq.|))))) is complex real ext-real Element of REAL
th2 is set
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
|.(bq).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
|.(bq).| . th2 is complex real ext-real Element of REAL
(bq) . th2 is complex set
|.((bq) . th2).| is complex real ext-real Element of REAL
Re ((bq) . th2) is complex real ext-real Element of REAL
(Re ((bq) . th2)) ^2 is complex real ext-real Element of REAL
(Re ((bq) . th2)) * (Re ((bq) . th2)) is complex real ext-real set
Im ((bq) . th2) is complex real ext-real Element of REAL
(Im ((bq) . th2)) ^2 is complex real ext-real Element of REAL
(Im ((bq) . th2)) * (Im ((bq) . th2)) is complex real ext-real set
((Re ((bq) . th2)) ^2) + ((Im ((bq) . th2)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . th2)) ^2) + ((Im ((bq) . th2)) ^2)) is complex real ext-real Element of REAL
(|.bq.|) . th2 is complex real ext-real Element of REAL
Partial_Sums (|.bq.|) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
PL is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th2 + PL is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(th2 + PL) + (th2 + PL) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₁₂ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq,bq,c₁₁) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
|.(bq,bq,c₁₁).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
|.(bq,bq,c₁₁).| . c₁₂ is complex real ext-real Element of REAL
(bq,bq,c₁₁) . c₁₂ is complex Element of COMPLEX
|.((bq,bq,c₁₁) . c₁₂).| is complex real ext-real Element of REAL
Re ((bq,bq,c₁₁) . c₁₂) is complex real ext-real Element of REAL
(Re ((bq,bq,c₁₁) . c₁₂)) ^2 is complex real ext-real Element of REAL
(Re ((bq,bq,c₁₁) . c₁₂)) * (Re ((bq,bq,c₁₁) . c₁₂)) is complex real ext-real set
Im ((bq,bq,c₁₁) . c₁₂) is complex real ext-real Element of REAL
(Im ((bq,bq,c₁₁) . c₁₂)) ^2 is complex real ext-real Element of REAL
(Im ((bq,bq,c₁₁) . c₁₂)) * (Im ((bq,bq,c₁₁) . c₁₂)) is complex real ext-real set
((Re ((bq,bq,c₁₁) . c₁₂)) ^2) + ((Im ((bq,bq,c₁₁) . c₁₂)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq,bq,c₁₁) . c₁₂)) ^2) + ((Im ((bq,bq,c₁₁) . c₁₂)) ^2)) is complex real ext-real Element of REAL
(bq) . c₁₂ is complex Element of COMPLEX
(Partial_Sums (bq)) . c₁₁ is complex Element of COMPLEX
c₁₁ -' c₁₂ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . (c₁₁ -' c₁₂) is complex Element of COMPLEX
((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)) is complex Element of COMPLEX
- ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)) is complex set
((Partial_Sums (bq)) . c₁₁) + (- ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))) is complex set
((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))) is complex Element of COMPLEX
|.(((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))).| is complex real ext-real Element of REAL
Re (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) is complex real ext-real Element of REAL
(Re (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))))) ^2 is complex real ext-real Element of REAL
(Re (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))))) * (Re (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))))) is complex real ext-real set
Im (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) is complex real ext-real Element of REAL
(Im (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))))) ^2 is complex real ext-real Element of REAL
(Im (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))))) * (Im (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))))) is complex real ext-real set
((Re (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))))) ^2) + ((Im (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))))) ^2) + ((Im (((bq) . c₁₂) * (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))))) ^2)) is complex real ext-real Element of REAL
|.((bq) . c₁₂).| is complex real ext-real Element of REAL
Re ((bq) . c₁₂) is complex real ext-real Element of REAL
(Re ((bq) . c₁₂)) ^2 is complex real ext-real Element of REAL
(Re ((bq) . c₁₂)) * (Re ((bq) . c₁₂)) is complex real ext-real set
Im ((bq) . c₁₂) is complex real ext-real Element of REAL
(Im ((bq) . c₁₂)) ^2 is complex real ext-real Element of REAL
(Im ((bq) . c₁₂)) * (Im ((bq) . c₁₂)) is complex real ext-real set
((Re ((bq) . c₁₂)) ^2) + ((Im ((bq) . c₁₂)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq) . c₁₂)) ^2) + ((Im ((bq) . c₁₂)) ^2)) is complex real ext-real Element of REAL
|.(((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))).| is complex real ext-real Element of REAL
Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) * (Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) is complex real ext-real set
Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) * (Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) is complex real ext-real set
((Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2) + ((Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2) + ((Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2)) is complex real ext-real Element of REAL
|.((bq) . c₁₂).| * |.(((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))).| is complex real ext-real Element of REAL
(|.bq.|) . c₁₂ is complex real ext-real Element of REAL
((|.bq.|) . c₁₂) * |.(((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))).| is complex real ext-real Element of REAL
c₁₂ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
|.(bq,bq,c₁₁).| . c₁₂ is complex real ext-real Element of REAL
(|.bq.|) . c₁₂ is complex real ext-real Element of REAL
c₁₁ -' c₁₂ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . (c₁₁ -' c₁₂) is complex Element of COMPLEX
((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)) is complex Element of COMPLEX
- ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)) is complex set
((Partial_Sums (bq)) . c₁₁) + (- ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))) is complex set
|.(((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))).| is complex real ext-real Element of REAL
Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) * (Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) is complex real ext-real set
Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) * (Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) is complex real ext-real set
((Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2) + ((Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2) + ((Im (((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂)))) ^2)) is complex real ext-real Element of REAL
((|.bq.|) . c₁₂) * |.(((Partial_Sums (bq)) . c₁₁) - ((Partial_Sums (bq)) . (c₁₁ -' c₁₂))).| is complex real ext-real Element of REAL
c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq,bq,k) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
|.(bq,bq,k).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
|.(bq,bq,k).| . c₁₁ is complex real ext-real Element of REAL
(|.bq.|) . c₁₁ is complex real ext-real Element of REAL
(Partial_Sums (bq)) . k is complex Element of COMPLEX
k -' c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . (k -' c₁₁) is complex Element of COMPLEX
((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)) is complex Element of COMPLEX
- ((Partial_Sums (bq)) . (k -' c₁₁)) is complex set
((Partial_Sums (bq)) . k) + (- ((Partial_Sums (bq)) . (k -' c₁₁))) is complex set
|.(((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁))).| is complex real ext-real Element of REAL
Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) * (Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) is complex real ext-real set
Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) * (Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) is complex real ext-real set
((Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2) + ((Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2) + ((Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2)) is complex real ext-real Element of REAL
((|.bq.|) . c₁₁) * |.(((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁))).| is complex real ext-real Element of REAL
|.((Partial_Sums (bq)) . k).| is complex real ext-real Element of REAL
Re ((Partial_Sums (bq)) . k) is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . k)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . k)) * (Re ((Partial_Sums (bq)) . k)) is complex real ext-real set
Im ((Partial_Sums (bq)) . k) is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . k)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . k)) * (Im ((Partial_Sums (bq)) . k)) is complex real ext-real set
((Re ((Partial_Sums (bq)) . k)) ^2) + ((Im ((Partial_Sums (bq)) . k)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums (bq)) . k)) ^2) + ((Im ((Partial_Sums (bq)) . k)) ^2)) is complex real ext-real Element of REAL
|.((Partial_Sums (bq)) . (k -' c₁₁)).| is complex real ext-real Element of REAL
Re ((Partial_Sums (bq)) . (k -' c₁₁)) is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . (k -' c₁₁))) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq)) . (k -' c₁₁))) * (Re ((Partial_Sums (bq)) . (k -' c₁₁))) is complex real ext-real set
Im ((Partial_Sums (bq)) . (k -' c₁₁)) is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . (k -' c₁₁))) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq)) . (k -' c₁₁))) * (Im ((Partial_Sums (bq)) . (k -' c₁₁))) is complex real ext-real set
((Re ((Partial_Sums (bq)) . (k -' c₁₁))) ^2) + ((Im ((Partial_Sums (bq)) . (k -' c₁₁))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums (bq)) . (k -' c₁₁))) ^2) + ((Im ((Partial_Sums (bq)) . (k -' c₁₁))) ^2)) is complex real ext-real Element of REAL
|.((Partial_Sums (bq)) . k).| + |.((Partial_Sums (bq)) . (k -' c₁₁)).| is complex real ext-real Element of REAL
(Sum (|.bq.|)) + |.((Partial_Sums (bq)) . (k -' c₁₁)).| is complex real ext-real Element of REAL
(Sum (|.bq.|)) + (Sum (|.bq.|)) is complex real ext-real Element of REAL
2 * (Sum (|.bq.|)) is complex real ext-real Element of REAL
((|.bq.|) . c₁₁) * (2 * (Sum (|.bq.|))) is complex real ext-real Element of REAL
c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
|.(bq,bq,k).| . c₁₁ is complex real ext-real Element of REAL
(|.bq.|) . c₁₁ is complex real ext-real Element of REAL
((|.bq.|) . c₁₁) * (2 * (Sum (|.bq.|))) is complex real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
0 + (th2 + PL) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th2 + 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
k -' c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
k - c₁₁ is complex real ext-real Element of REAL
- c₁₁ is complex real ext-real non positive set
k + (- c₁₁) is complex real ext-real set
0 + PL is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
p - c₁₁ is complex real ext-real Element of REAL
p + (- c₁₁) is complex real ext-real set
((th2 + PL) + (th2 + PL)) - (th2 + PL) is complex real ext-real Element of REAL
- (th2 + PL) is complex real ext-real non positive set
((th2 + PL) + (th2 + PL)) + (- (th2 + PL)) is complex real ext-real set
((th2 + PL) + (th2 + PL)) - c₁₁ is complex real ext-real Element of REAL
((th2 + PL) + (th2 + PL)) + (- c₁₁) is complex real ext-real set
(Partial_Sums (bq)) . k is complex Element of COMPLEX
(Partial_Sums (bq)) . (k -' c₁₁) is complex Element of COMPLEX
((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)) is complex Element of COMPLEX
- ((Partial_Sums (bq)) . (k -' c₁₁)) is complex set
((Partial_Sums (bq)) . k) + (- ((Partial_Sums (bq)) . (k -' c₁₁))) is complex set
|.(((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁))).| is complex real ext-real Element of REAL
Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) * (Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) is complex real ext-real set
Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) * (Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) is complex real ext-real set
((Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2) + ((Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2) + ((Im (((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁)))) ^2)) is complex real ext-real Element of REAL
(|.bq.|) . c₁₁ is complex real ext-real Element of REAL
((|.bq.|) . c₁₁) * |.(((Partial_Sums (bq)) . k) - ((Partial_Sums (bq)) . (k -' c₁₁))).| is complex real ext-real Element of REAL
((|.bq.|) . c₁₁) * (min ((b / (4 * (Sum (|.bq.|)))),(b / (4 * (Sum (|.bq.|)))))) is complex real ext-real Element of REAL
(bq,bq,k) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
|.(bq,bq,k).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
|.(bq,bq,k).| . c₁₁ is complex real ext-real Element of REAL
(min ((b / (4 * (Sum (|.bq.|)))),(b / (4 * (Sum (|.bq.|)))))) * ((|.bq.|) . c₁₁) is complex real ext-real Element of REAL
Partial_Sums |.(bq,bq,k).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued non-decreasing Element of K19(K20(NAT,REAL))
(Partial_Sums |.(bq,bq,k).|) . (th2 + PL) is complex real ext-real Element of REAL
(Partial_Sums (|.bq.|)) . (th2 + PL) is complex real ext-real Element of REAL
((Partial_Sums (|.bq.|)) . (th2 + PL)) * (min ((b / (4 * (Sum (|.bq.|)))),(b / (4 * (Sum (|.bq.|)))))) is complex real ext-real Element of REAL
(Sum (|.bq.|)) * (min ((b / (4 * (Sum (|.bq.|)))),(b / (4 * (Sum (|.bq.|)))))) is complex real ext-real Element of REAL
a / (4 * (Sum (|.bq.|))) is complex real ext-real Element of REAL
a * ((4 * (Sum (|.bq.|))) ") is complex real ext-real set
(Sum (|.bq.|)) * (a / (4 * (Sum (|.bq.|)))) is complex real ext-real Element of REAL
(Sum (|.bq.|)) * a is complex real ext-real Element of REAL
((Sum (|.bq.|)) * a) / (4 * (Sum (|.bq.|))) is complex real ext-real Element of REAL
((Sum (|.bq.|)) * a) * ((4 * (Sum (|.bq.|))) ") is complex real ext-real set
a / 4 is complex real ext-real Element of REAL
4 " is non empty complex real ext-real positive non negative set
a * (4 ") is complex real ext-real set
(a / 4) + (a / 4) is complex real ext-real Element of REAL
0 + (a / 4) is complex real ext-real Element of REAL
a / 2 is complex real ext-real Element of REAL
2 " is non empty complex real ext-real positive non negative set
a * (2 ") is complex real ext-real set
k -' (th2 + PL) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
k - (th2 + PL) is complex real ext-real Element of REAL
k + (- (th2 + PL)) is complex real ext-real set
(k -' (th2 + PL)) + (th2 + PL) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
|.(bq,bq,k).| . c₁₁ is complex real ext-real Element of REAL
(|.bq.|) . c₁₁ is complex real ext-real Element of REAL
(2 * (Sum (|.bq.|))) * ((|.bq.|) . c₁₁) is complex real ext-real Element of REAL
(Partial_Sums |.(bq,bq,k).|) . k is complex real ext-real Element of REAL
((Partial_Sums |.(bq,bq,k).|) . k) - ((Partial_Sums |.(bq,bq,k).|) . (th2 + PL)) is complex real ext-real Element of REAL
- ((Partial_Sums |.(bq,bq,k).|) . (th2 + PL)) is complex real ext-real set
((Partial_Sums |.(bq,bq,k).|) . k) + (- ((Partial_Sums |.(bq,bq,k).|) . (th2 + PL))) is complex real ext-real set
(Partial_Sums (|.bq.|)) . k is complex real ext-real Element of REAL
((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)) is complex real ext-real Element of REAL
- ((Partial_Sums (|.bq.|)) . (th2 + PL)) is complex real ext-real set
((Partial_Sums (|.bq.|)) . k) + (- ((Partial_Sums (|.bq.|)) . (th2 + PL))) is complex real ext-real set
(((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL))) * (2 * (Sum (|.bq.|))) is complex real ext-real Element of REAL
abs (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL))) is complex real ext-real Element of REAL
Re (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)))) * (Re (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)))) is complex real ext-real set
Im (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)))) * (Im (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)))) is complex real ext-real set
((Re (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)))) ^2) + ((Im (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)))) ^2) + ((Im (((Partial_Sums (|.bq.|)) . k) - ((Partial_Sums (|.bq.|)) . (th2 + PL)))) ^2)) is complex real ext-real Element of REAL
(min ((b / (4 * (Sum (|.bq.|)))),(b / (4 * (Sum (|.bq.|)))))) * (2 * (Sum (|.bq.|))) is complex real ext-real Element of REAL
(2 * (Sum (|.bq.|))) * (min ((b / (4 * (Sum (|.bq.|)))),(b / (4 * (Sum (|.bq.|)))))) is complex real ext-real Element of REAL
a / (4 * (Sum (|.bq.|))) is complex real ext-real Element of REAL
a * ((4 * (Sum (|.bq.|))) ") is complex real ext-real set
(2 * (Sum (|.bq.|))) * (a / (4 * (Sum (|.bq.|)))) is complex real ext-real Element of REAL
2 * a is complex real ext-real Element of REAL
(2 * a) * (Sum (|.bq.|)) is complex real ext-real Element of REAL
((2 * a) * (Sum (|.bq.|))) / (4 * (Sum (|.bq.|))) is complex real ext-real Element of REAL
((2 * a) * (Sum (|.bq.|))) * ((4 * (Sum (|.bq.|))) ") is complex real ext-real set
a + a is complex real ext-real set
(a + a) / 4 is complex real ext-real Element of REAL
(a + a) * (4 ") is complex real ext-real set
(a / 2) + (a / 2) is complex real ext-real Element of REAL
(((Partial_Sums |.(bq,bq,k).|) . k) - ((Partial_Sums |.(bq,bq,k).|) . (th2 + PL))) + ((Partial_Sums |.(bq,bq,k).|) . (th2 + PL)) is complex real ext-real Element of REAL
abs ((Partial_Sums |.(bq,bq,k).|) . k) is complex real ext-real Element of REAL
Re ((Partial_Sums |.(bq,bq,k).|) . k) is complex real ext-real Element of REAL
(Re ((Partial_Sums |.(bq,bq,k).|) . k)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums |.(bq,bq,k).|) . k)) * (Re ((Partial_Sums |.(bq,bq,k).|) . k)) is complex real ext-real set
Im ((Partial_Sums |.(bq,bq,k).|) . k) is complex real ext-real Element of REAL
(Im ((Partial_Sums |.(bq,bq,k).|) . k)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums |.(bq,bq,k).|) . k)) * (Im ((Partial_Sums |.(bq,bq,k).|) . k)) is complex real ext-real set
((Re ((Partial_Sums |.(bq,bq,k).|) . k)) ^2) + ((Im ((Partial_Sums |.(bq,bq,k).|) . k)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums |.(bq,bq,k).|) . k)) ^2) + ((Im ((Partial_Sums |.(bq,bq,k).|) . k)) ^2)) is complex real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq,bq,k) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
|.(bq,bq,k).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums |.(bq,bq,k).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued non-decreasing Element of K19(K20(NAT,REAL))
(Partial_Sums |.(bq,bq,k).|) . k is complex real ext-real Element of REAL
abs ((Partial_Sums |.(bq,bq,k).|) . k) is complex real ext-real Element of REAL
Re ((Partial_Sums |.(bq,bq,k).|) . k) is complex real ext-real Element of REAL
(Re ((Partial_Sums |.(bq,bq,k).|) . k)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums |.(bq,bq,k).|) . k)) * (Re ((Partial_Sums |.(bq,bq,k).|) . k)) is complex real ext-real set
Im ((Partial_Sums |.(bq,bq,k).|) . k) is complex real ext-real Element of REAL
(Im ((Partial_Sums |.(bq,bq,k).|) . k)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums |.(bq,bq,k).|) . k)) * (Im ((Partial_Sums |.(bq,bq,k).|) . k)) is complex real ext-real set
((Re ((Partial_Sums |.(bq,bq,k).|) . k)) ^2) + ((Im ((Partial_Sums |.(bq,bq,k).|) . k)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums |.(bq,bq,k).|) . k)) ^2) + ((Im ((Partial_Sums |.(bq,bq,k).|) . k)) ^2)) is complex real ext-real Element of REAL
bq is complex Element of COMPLEX
bq is complex Element of COMPLEX
a is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim a is complex Element of COMPLEX
b is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th1 is complex Element of COMPLEX
|.(b . th1).| is complex real ext-real Element of REAL
Re (b . th1) is complex real ext-real Element of REAL
(Re (b . th1)) ^2 is complex real ext-real Element of REAL
(Re (b . th1)) * (Re (b . th1)) is complex real ext-real set
Im (b . th1) is complex real ext-real Element of REAL
(Im (b . th1)) ^2 is complex real ext-real Element of REAL
(Im (b . th1)) * (Im (b . th1)) is complex real ext-real set
((Re (b . th1)) ^2) + ((Im (b . th1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (b . th1)) ^2) + ((Im (b . th1)) ^2)) is complex real ext-real Element of REAL
(bq,bq,th1) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (bq,bq,th1) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (bq,bq,th1)) . th1 is complex Element of COMPLEX
|.((Partial_Sums (bq,bq,th1)) . th1).| is complex real ext-real Element of REAL
Re ((Partial_Sums (bq,bq,th1)) . th1) is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq,bq,th1)) . th1)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums (bq,bq,th1)) . th1)) * (Re ((Partial_Sums (bq,bq,th1)) . th1)) is complex real ext-real set
Im ((Partial_Sums (bq,bq,th1)) . th1) is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq,bq,th1)) . th1)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums (bq,bq,th1)) . th1)) * (Im ((Partial_Sums (bq,bq,th1)) . th1)) is complex real ext-real set
((Re ((Partial_Sums (bq,bq,th1)) . th1)) ^2) + ((Im ((Partial_Sums (bq,bq,th1)) . th1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums (bq,bq,th1)) . th1)) ^2) + ((Im ((Partial_Sums (bq,bq,th1)) . th1)) ^2)) is complex real ext-real Element of REAL
|.(bq,bq,th1).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums |.(bq,bq,th1).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued non-decreasing Element of K19(K20(NAT,REAL))
(Partial_Sums |.(bq,bq,th1).|) . th1 is complex real ext-real Element of REAL
th1 is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
th1 is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th2 is complex Element of COMPLEX
|.(b . th2).| is complex real ext-real Element of REAL
Re (b . th2) is complex real ext-real Element of REAL
(Re (b . th2)) ^2 is complex real ext-real Element of REAL
(Re (b . th2)) * (Re (b . th2)) is complex real ext-real set
Im (b . th2) is complex real ext-real Element of REAL
(Im (b . th2)) ^2 is complex real ext-real Element of REAL
(Im (b . th2)) * (Im (b . th2)) is complex real ext-real set
((Re (b . th2)) ^2) + ((Im (b . th2)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (b . th2)) ^2) + ((Im (b . th2)) ^2)) is complex real ext-real Element of REAL
(bq,bq,th2) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
|.(bq,bq,th2).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums |.(bq,bq,th2).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued non-decreasing Element of K19(K20(NAT,REAL))
(Partial_Sums |.(bq,bq,th2).|) . th2 is complex real ext-real Element of REAL
th1 . th2 is complex real ext-real Element of REAL
th2 is complex real ext-real set
PL is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₈ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 . p is complex real ext-real Element of REAL
(th1 . p) - 0 is complex real ext-real Element of REAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
(th1 . p) + (- 0) is complex real ext-real set
abs ((th1 . p) - 0) is complex real ext-real Element of REAL
Re ((th1 . p) - 0) is complex real ext-real Element of REAL
(Re ((th1 . p) - 0)) ^2 is complex real ext-real Element of REAL
(Re ((th1 . p) - 0)) * (Re ((th1 . p) - 0)) is complex real ext-real set
Im ((th1 . p) - 0) is complex real ext-real Element of REAL
(Im ((th1 . p) - 0)) ^2 is complex real ext-real Element of REAL
(Im ((th1 . p) - 0)) * (Im ((th1 . p) - 0)) is complex real ext-real set
((Re ((th1 . p) - 0)) ^2) + ((Im ((th1 . p) - 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((th1 . p) - 0)) ^2) + ((Im ((th1 . p) - 0)) ^2)) is complex real ext-real Element of REAL
(bq,bq,p) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
|.(bq,bq,p).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums |.(bq,bq,p).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued non-decreasing Element of K19(K20(NAT,REAL))
(Partial_Sums |.(bq,bq,p).|) . p is complex real ext-real Element of REAL
abs ((Partial_Sums |.(bq,bq,p).|) . p) is complex real ext-real Element of REAL
Re ((Partial_Sums |.(bq,bq,p).|) . p) is complex real ext-real Element of REAL
(Re ((Partial_Sums |.(bq,bq,p).|) . p)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums |.(bq,bq,p).|) . p)) * (Re ((Partial_Sums |.(bq,bq,p).|) . p)) is complex real ext-real set
Im ((Partial_Sums |.(bq,bq,p).|) . p) is complex real ext-real Element of REAL
(Im ((Partial_Sums |.(bq,bq,p).|) . p)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums |.(bq,bq,p).|) . p)) * (Im ((Partial_Sums |.(bq,bq,p).|) . p)) is complex real ext-real set
((Re ((Partial_Sums |.(bq,bq,p).|) . p)) ^2) + ((Im ((Partial_Sums |.(bq,bq,p).|) . p)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums |.(bq,bq,p).|) . p)) ^2) + ((Im ((Partial_Sums |.(bq,bq,p).|) . p)) ^2)) is complex real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 . p is complex real ext-real Element of REAL
(th1 . p) - 0 is complex real ext-real Element of REAL
(th1 . p) + (- 0) is complex real ext-real set
abs ((th1 . p) - 0) is complex real ext-real Element of REAL
Re ((th1 . p) - 0) is complex real ext-real Element of REAL
(Re ((th1 . p) - 0)) ^2 is complex real ext-real Element of REAL
(Re ((th1 . p) - 0)) * (Re ((th1 . p) - 0)) is complex real ext-real set
Im ((th1 . p) - 0) is complex real ext-real Element of REAL
(Im ((th1 . p) - 0)) ^2 is complex real ext-real Element of REAL
(Im ((th1 . p) - 0)) * (Im ((th1 . p) - 0)) is complex real ext-real set
((Re ((th1 . p) - 0)) ^2) + ((Im ((th1 . p) - 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((th1 . p) - 0)) ^2) + ((Im ((th1 . p) - 0)) ^2)) is complex real ext-real Element of REAL
lim th1 is complex real ext-real Element of REAL
lim b is complex Element of COMPLEX
bq is complex set
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (bq) is complex Element of COMPLEX
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (bq)) is complex Element of COMPLEX
bq is complex set
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (bq) is complex Element of COMPLEX
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (bq)) is complex Element of COMPLEX
(Sum (bq)) * (Sum (bq)) is complex Element of COMPLEX
bq + bq is complex set
((bq + bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq + bq)) is complex Element of COMPLEX
Partial_Sums ((bq + bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq + bq))) is complex Element of COMPLEX
a is complex Element of COMPLEX
b is complex Element of COMPLEX
th1 is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th1 is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 . th2 is complex Element of COMPLEX
(a,b,th2) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Partial_Sums (a,b,th2) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (a,b,th2)) . th2 is complex Element of COMPLEX
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (a)) . th2 is complex Element of COMPLEX
(b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums (b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
(Partial_Sums (b)) . th2 is complex Element of COMPLEX
((Partial_Sums (a)) . th2) * ((Partial_Sums (b)) . th2) is complex Element of COMPLEX
a + b is complex Element of COMPLEX
((a + b)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Partial_Sums ((a + b)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
(Partial_Sums ((a + b))) . th2 is complex Element of COMPLEX
(((Partial_Sums (a)) . th2) * ((Partial_Sums (b)) . th2)) - ((Partial_Sums ((a + b))) . th2) is complex Element of COMPLEX
- ((Partial_Sums ((a + b))) . th2) is complex set
(((Partial_Sums (a)) . th2) * ((Partial_Sums (b)) . th2)) + (- ((Partial_Sums ((a + b))) . th2)) is complex set
(Partial_Sums (a)) (#) (Partial_Sums (b)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
((Partial_Sums (a)) (#) (Partial_Sums (b))) . th2 is complex Element of COMPLEX
(((Partial_Sums (a)) (#) (Partial_Sums (b))) . th2) - ((Partial_Sums ((a + b))) . th2) is complex Element of COMPLEX
(((Partial_Sums (a)) (#) (Partial_Sums (b))) . th2) + (- ((Partial_Sums ((a + b))) . th2)) is complex set
- (Partial_Sums ((a + b))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
(- 1) (#) (Partial_Sums ((a + b))) is Relation-like NAT -defined Function-like total complex-valued set
(- (Partial_Sums ((a + b)))) . th2 is complex Element of COMPLEX
(((Partial_Sums (a)) (#) (Partial_Sums (b))) . th2) + ((- (Partial_Sums ((a + b)))) . th2) is complex Element of COMPLEX
((Partial_Sums (a)) (#) (Partial_Sums (b))) - (Partial_Sums ((a + b))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
- (Partial_Sums ((a + b))) is Relation-like NAT -defined Function-like total complex-valued set
((Partial_Sums (a)) (#) (Partial_Sums (b))) + (- (Partial_Sums ((a + b)))) is Relation-like NAT -defined Function-like total complex-valued set
(((Partial_Sums (a)) (#) (Partial_Sums (b))) - (Partial_Sums ((a + b)))) . th2 is complex Element of COMPLEX
lim ((Partial_Sums (a)) (#) (Partial_Sums (b))) is complex Element of COMPLEX
lim (Partial_Sums (a)) is complex Element of COMPLEX
lim (Partial_Sums (b)) is complex Element of COMPLEX
(lim (Partial_Sums (a))) * (lim (Partial_Sums (b))) is complex Element of COMPLEX
lim th1 is complex Element of COMPLEX
bq is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(COMPLEX,COMPLEX))
bq is complex set
bq . bq is complex set
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (bq) is complex Element of COMPLEX
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (bq)) is complex Element of COMPLEX
bq is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(COMPLEX,COMPLEX))
bq is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(COMPLEX,COMPLEX))
a is complex Element of COMPLEX
bq . a is complex Element of COMPLEX
bq . a is complex Element of COMPLEX
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
() is Relation-like COMPLEX -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(COMPLEX,COMPLEX))
bq is complex set
() . bq is complex set
bq is complex set
(bq) is complex set
() . bq is complex set
bq is complex set
bq is complex set
bq + bq is complex set
((bq + bq)) is complex Element of COMPLEX
() . (bq + bq) is complex set
(bq) is complex Element of COMPLEX
() . bq is complex set
(bq) is complex Element of COMPLEX
() . bq is complex set
(bq) * (bq) is complex Element of COMPLEX
((bq + bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq + bq)) is complex Element of COMPLEX
Partial_Sums ((bq + bq)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq + bq))) is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (bq) is complex Element of COMPLEX
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (bq)) is complex Element of COMPLEX
(bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (bq) is complex Element of COMPLEX
Partial_Sums (bq) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (bq)) is complex Element of COMPLEX
(Sum (bq)) * (Sum (bq)) is complex Element of COMPLEX
(bq) * (Sum (bq)) is complex Element of COMPLEX
bq is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
bq is complex real ext-real Element of REAL
bq . bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
Im (Sum ((bq * <i>))) is complex real ext-real Element of REAL
bq is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
bq is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
a is complex real ext-real Element of REAL
bq . a is complex real ext-real Element of REAL
bq . a is complex real ext-real Element of REAL
a * <i> is complex set
((a * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((a * <i>)) is complex Element of COMPLEX
Partial_Sums ((a * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((a * <i>))) is complex Element of COMPLEX
Im (Sum ((a * <i>))) is complex real ext-real Element of REAL
() is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
bq is complex real ext-real set
() . bq is complex real ext-real Element of REAL
bq is complex real ext-real set
(bq) is set
() . bq is complex real ext-real Element of REAL
bq is complex real ext-real Element of REAL
(bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
bq is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
bq is complex real ext-real Element of REAL
bq . bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
Re (Sum ((bq * <i>))) is complex real ext-real Element of REAL
bq is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
bq is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
a is complex real ext-real Element of REAL
bq . a is complex real ext-real Element of REAL
bq . a is complex real ext-real Element of REAL
a * <i> is complex set
((a * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((a * <i>)) is complex Element of COMPLEX
Partial_Sums ((a * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((a * <i>))) is complex Element of COMPLEX
Re (Sum ((a * <i>))) is complex real ext-real Element of REAL
() is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
bq is complex real ext-real set
() . bq is complex real ext-real Element of REAL
bq is complex real ext-real set
(bq) is set
() . bq is complex real ext-real Element of REAL
bq is complex real ext-real Element of REAL
(bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
dom () is V46() V47() V48() Element of K19(REAL)
dom () is V46() V47() V48() Element of K19(REAL)
bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
() . bq is complex real ext-real Element of REAL
() . bq is complex real ext-real Element of REAL
(() . bq) * <i> is complex set
(() . bq) + ((() . bq) * <i>) is complex set
Im (Sum ((bq * <i>))) is complex real ext-real Element of REAL
Re (Sum ((bq * <i>))) is complex real ext-real Element of REAL
bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is complex Element of COMPLEX
() . (bq * <i>) is complex set
(bq) is complex real ext-real Element of REAL
() . bq is complex real ext-real Element of REAL
(bq) is complex real ext-real Element of REAL
() . bq is complex real ext-real Element of REAL
(bq) * <i> is complex set
(bq) + ((bq) * <i>) is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
(Sum ((bq * <i>))) *' is complex Element of COMPLEX
Re (Sum ((bq * <i>))) is complex real ext-real Element of REAL
Im (Sum ((bq * <i>))) is complex real ext-real Element of REAL
(Im (Sum ((bq * <i>)))) * <i> is complex set
(Re (Sum ((bq * <i>)))) - ((Im (Sum ((bq * <i>)))) * <i>) is complex set
- ((Im (Sum ((bq * <i>)))) * <i>) is complex set
(Re (Sum ((bq * <i>)))) + (- ((Im (Sum ((bq * <i>)))) * <i>)) is complex set
- (bq * <i>) is complex set
((- (bq * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((- (bq * <i>))) is complex Element of COMPLEX
Partial_Sums ((- (bq * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((- (bq * <i>)))) is complex Element of COMPLEX
(Partial_Sums ((bq * <i>))) *' is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
((bq * <i>)) . 0 is complex Element of COMPLEX
(((bq * <i>)) . 0) *' is complex Element of COMPLEX
Re (((bq * <i>)) . 0) is complex real ext-real Element of REAL
Im (((bq * <i>)) . 0) is complex real ext-real Element of REAL
(Im (((bq * <i>)) . 0)) * <i> is complex set
(Re (((bq * <i>)) . 0)) - ((Im (((bq * <i>)) . 0)) * <i>) is complex set
- ((Im (((bq * <i>)) . 0)) * <i>) is complex set
(Re (((bq * <i>)) . 0)) + (- ((Im (((bq * <i>)) . 0)) * <i>)) is complex set
((- (bq * <i>))) . 0 is complex Element of COMPLEX
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((bq * <i>)) . bq is complex Element of COMPLEX
(((bq * <i>)) . bq) *' is complex Element of COMPLEX
Re (((bq * <i>)) . bq) is complex real ext-real Element of REAL
Im (((bq * <i>)) . bq) is complex real ext-real Element of REAL
(Im (((bq * <i>)) . bq)) * <i> is complex set
(Re (((bq * <i>)) . bq)) - ((Im (((bq * <i>)) . bq)) * <i>) is complex set
- ((Im (((bq * <i>)) . bq)) * <i>) is complex set
(Re (((bq * <i>)) . bq)) + (- ((Im (((bq * <i>)) . bq)) * <i>)) is complex set
((- (bq * <i>))) . bq is complex Element of COMPLEX
bq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((bq * <i>)) . (bq + 1) is complex Element of COMPLEX
(((bq * <i>)) . (bq + 1)) *' is complex Element of COMPLEX
Re (((bq * <i>)) . (bq + 1)) is complex real ext-real Element of REAL
Im (((bq * <i>)) . (bq + 1)) is complex real ext-real Element of REAL
(Im (((bq * <i>)) . (bq + 1))) * <i> is complex set
(Re (((bq * <i>)) . (bq + 1))) - ((Im (((bq * <i>)) . (bq + 1))) * <i>) is complex set
- ((Im (((bq * <i>)) . (bq + 1))) * <i>) is complex set
(Re (((bq * <i>)) . (bq + 1))) + (- ((Im (((bq * <i>)) . (bq + 1))) * <i>)) is complex set
((- (bq * <i>))) . (bq + 1) is complex Element of COMPLEX
(((bq * <i>)) . bq) * (bq * <i>) is complex set
(bq + 1) + (0 * <i>) is complex set
((((bq * <i>)) . bq) * (bq * <i>)) / ((bq + 1) + (0 * <i>)) is complex Element of COMPLEX
((bq + 1) + (0 * <i>)) " is complex set
((((bq * <i>)) . bq) * (bq * <i>)) * (((bq + 1) + (0 * <i>)) ") is complex set
(((((bq * <i>)) . bq) * (bq * <i>)) / ((bq + 1) + (0 * <i>))) *' is complex Element of COMPLEX
Re (((((bq * <i>)) . bq) * (bq * <i>)) / ((bq + 1) + (0 * <i>))) is complex real ext-real Element of REAL
Im (((((bq * <i>)) . bq) * (bq * <i>)) / ((bq + 1) + (0 * <i>))) is complex real ext-real Element of REAL
(Im (((((bq * <i>)) . bq) * (bq * <i>)) / ((bq + 1) + (0 * <i>)))) * <i> is complex set
(Re (((((bq * <i>)) . bq) * (bq * <i>)) / ((bq + 1) + (0 * <i>)))) - ((Im (((((bq * <i>)) . bq) * (bq * <i>)) / ((bq + 1) + (0 * <i>)))) * <i>) is complex set
- ((Im (((((bq * <i>)) . bq) * (bq * <i>)) / ((bq + 1) + (0 * <i>)))) * <i>) is complex set
(Re (((((bq * <i>)) . bq) * (bq * <i>)) / ((bq + 1) + (0 * <i>)))) + (- ((Im (((((bq * <i>)) . bq) * (bq * <i>)) / ((bq + 1) + (0 * <i>)))) * <i>)) is complex set
(bq * <i>) / ((bq + 1) + (0 * <i>)) is complex Element of COMPLEX
(bq * <i>) * (((bq + 1) + (0 * <i>)) ") is complex set
(((bq * <i>)) . bq) * ((bq * <i>) / ((bq + 1) + (0 * <i>))) is complex Element of COMPLEX
((((bq * <i>)) . bq) * ((bq * <i>) / ((bq + 1) + (0 * <i>)))) *' is complex Element of COMPLEX
Re ((((bq * <i>)) . bq) * ((bq * <i>) / ((bq + 1) + (0 * <i>)))) is complex real ext-real Element of REAL
Im ((((bq * <i>)) . bq) * ((bq * <i>) / ((bq + 1) + (0 * <i>)))) is complex real ext-real Element of REAL
(Im ((((bq * <i>)) . bq) * ((bq * <i>) / ((bq + 1) + (0 * <i>))))) * <i> is complex set
(Re ((((bq * <i>)) . bq) * ((bq * <i>) / ((bq + 1) + (0 * <i>))))) - ((Im ((((bq * <i>)) . bq) * ((bq * <i>) / ((bq + 1) + (0 * <i>))))) * <i>) is complex set
- ((Im ((((bq * <i>)) . bq) * ((bq * <i>) / ((bq + 1) + (0 * <i>))))) * <i>) is complex set
(Re ((((bq * <i>)) . bq) * ((bq * <i>) / ((bq + 1) + (0 * <i>))))) + (- ((Im ((((bq * <i>)) . bq) * ((bq * <i>) / ((bq + 1) + (0 * <i>))))) * <i>)) is complex set
(bq * <i>) / (bq + 1) is complex Element of COMPLEX
(bq + 1) " is non empty complex real ext-real positive non negative set
(bq * <i>) * ((bq + 1) ") is complex set
((bq * <i>) / (bq + 1)) *' is complex Element of COMPLEX
Re ((bq * <i>) / (bq + 1)) is complex real ext-real Element of REAL
Im ((bq * <i>) / (bq + 1)) is complex real ext-real Element of REAL
(Im ((bq * <i>) / (bq + 1))) * <i> is complex set
(Re ((bq * <i>) / (bq + 1))) - ((Im ((bq * <i>) / (bq + 1))) * <i>) is complex set
- ((Im ((bq * <i>) / (bq + 1))) * <i>) is complex set
(Re ((bq * <i>) / (bq + 1))) + (- ((Im ((bq * <i>) / (bq + 1))) * <i>)) is complex set
(((- (bq * <i>))) . bq) * (((bq * <i>) / (bq + 1)) *') is complex Element of COMPLEX
0 + (bq * <i>) is complex set
(0 + (bq * <i>)) *' is complex Element of COMPLEX
Re (0 + (bq * <i>)) is complex real ext-real Element of REAL
Im (0 + (bq * <i>)) is complex real ext-real Element of REAL
(Im (0 + (bq * <i>))) * <i> is complex set
(Re (0 + (bq * <i>))) - ((Im (0 + (bq * <i>))) * <i>) is complex set
- ((Im (0 + (bq * <i>))) * <i>) is complex set
(Re (0 + (bq * <i>))) + (- ((Im (0 + (bq * <i>))) * <i>)) is complex set
(bq + 1) *' is complex Element of COMPLEX
Re (bq + 1) is complex real ext-real Element of REAL
Im (bq + 1) is complex real ext-real Element of REAL
(Im (bq + 1)) * <i> is complex set
(Re (bq + 1)) - ((Im (bq + 1)) * <i>) is complex set
- ((Im (bq + 1)) * <i>) is complex set
(Re (bq + 1)) + (- ((Im (bq + 1)) * <i>)) is complex set
((0 + (bq * <i>)) *') / ((bq + 1) *') is complex Element of COMPLEX
((bq + 1) *') " is complex set
Re ((bq + 1) *') is complex real ext-real Element of REAL
(Re ((bq + 1) *')) ^2 is complex real ext-real Element of REAL
(Re ((bq + 1) *')) * (Re ((bq + 1) *')) is complex real ext-real set
Im ((bq + 1) *') is complex real ext-real Element of REAL
(Im ((bq + 1) *')) ^2 is complex real ext-real Element of REAL
(Im ((bq + 1) *')) * (Im ((bq + 1) *')) is complex real ext-real set
((Re ((bq + 1) *')) ^2) + ((Im ((bq + 1) *')) ^2) is complex real ext-real Element of REAL
(Re ((bq + 1) *')) / (((Re ((bq + 1) *')) ^2) + ((Im ((bq + 1) *')) ^2)) is complex real ext-real Element of REAL
- (Im ((bq + 1) *')) is complex real ext-real Element of REAL
(- (Im ((bq + 1) *'))) / (((Re ((bq + 1) *')) ^2) + ((Im ((bq + 1) *')) ^2)) is complex real ext-real Element of REAL
((- (Im ((bq + 1) *'))) / (((Re ((bq + 1) *')) ^2) + ((Im ((bq + 1) *')) ^2))) * <i> is complex set
((Re ((bq + 1) *')) / (((Re ((bq + 1) *')) ^2) + ((Im ((bq + 1) *')) ^2))) + (((- (Im ((bq + 1) *'))) / (((Re ((bq + 1) *')) ^2) + ((Im ((bq + 1) *')) ^2))) * <i>) is complex set
((0 + (bq * <i>)) *') * (((bq + 1) *') ") is complex set
(((- (bq * <i>))) . bq) * (((0 + (bq * <i>)) *') / ((bq + 1) *')) is complex Element of COMPLEX
- bq is complex real ext-real Element of REAL
(- bq) * <i> is complex set
0 + ((- bq) * <i>) is complex set
((bq + 1) + (0 * <i>)) *' is complex Element of COMPLEX
Re ((bq + 1) + (0 * <i>)) is complex real ext-real Element of REAL
Im ((bq + 1) + (0 * <i>)) is complex real ext-real Element of REAL
(Im ((bq + 1) + (0 * <i>))) * <i> is complex set
(Re ((bq + 1) + (0 * <i>))) - ((Im ((bq + 1) + (0 * <i>))) * <i>) is complex set
- ((Im ((bq + 1) + (0 * <i>))) * <i>) is complex set
(Re ((bq + 1) + (0 * <i>))) + (- ((Im ((bq + 1) + (0 * <i>))) * <i>)) is complex set
(0 + ((- bq) * <i>)) / (((bq + 1) + (0 * <i>)) *') is complex Element of COMPLEX
(((bq + 1) + (0 * <i>)) *') " is complex set
Re (((bq + 1) + (0 * <i>)) *') is complex real ext-real Element of REAL
(Re (((bq + 1) + (0 * <i>)) *')) ^2 is complex real ext-real Element of REAL
(Re (((bq + 1) + (0 * <i>)) *')) * (Re (((bq + 1) + (0 * <i>)) *')) is complex real ext-real set
Im (((bq + 1) + (0 * <i>)) *') is complex real ext-real Element of REAL
(Im (((bq + 1) + (0 * <i>)) *')) ^2 is complex real ext-real Element of REAL
(Im (((bq + 1) + (0 * <i>)) *')) * (Im (((bq + 1) + (0 * <i>)) *')) is complex real ext-real set
((Re (((bq + 1) + (0 * <i>)) *')) ^2) + ((Im (((bq + 1) + (0 * <i>)) *')) ^2) is complex real ext-real Element of REAL
(Re (((bq + 1) + (0 * <i>)) *')) / (((Re (((bq + 1) + (0 * <i>)) *')) ^2) + ((Im (((bq + 1) + (0 * <i>)) *')) ^2)) is complex real ext-real Element of REAL
- (Im (((bq + 1) + (0 * <i>)) *')) is complex real ext-real Element of REAL
(- (Im (((bq + 1) + (0 * <i>)) *'))) / (((Re (((bq + 1) + (0 * <i>)) *')) ^2) + ((Im (((bq + 1) + (0 * <i>)) *')) ^2)) is complex real ext-real Element of REAL
((- (Im (((bq + 1) + (0 * <i>)) *'))) / (((Re (((bq + 1) + (0 * <i>)) *')) ^2) + ((Im (((bq + 1) + (0 * <i>)) *')) ^2))) * <i> is complex set
((Re (((bq + 1) + (0 * <i>)) *')) / (((Re (((bq + 1) + (0 * <i>)) *')) ^2) + ((Im (((bq + 1) + (0 * <i>)) *')) ^2))) + (((- (Im (((bq + 1) + (0 * <i>)) *'))) / (((Re (((bq + 1) + (0 * <i>)) *')) ^2) + ((Im (((bq + 1) + (0 * <i>)) *')) ^2))) * <i>) is complex set
(0 + ((- bq) * <i>)) * ((((bq + 1) + (0 * <i>)) *') ") is complex set
(((- (bq * <i>))) . bq) * ((0 + ((- bq) * <i>)) / (((bq + 1) + (0 * <i>)) *')) is complex Element of COMPLEX
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of REAL
(- 0) * <i> is complex set
(bq + 1) + ((- 0) * <i>) is complex set
(0 + ((- bq) * <i>)) / ((bq + 1) + ((- 0) * <i>)) is complex Element of COMPLEX
((bq + 1) + ((- 0) * <i>)) " is complex set
(0 + ((- bq) * <i>)) * (((bq + 1) + ((- 0) * <i>)) ") is complex set
(((- (bq * <i>))) . bq) * ((0 + ((- bq) * <i>)) / ((bq + 1) + ((- 0) * <i>))) is complex Element of COMPLEX
(((- (bq * <i>))) . bq) * (- (bq * <i>)) is complex set
((((- (bq * <i>))) . bq) * (- (bq * <i>))) / ((bq + 1) + (0 * <i>)) is complex Element of COMPLEX
((((- (bq * <i>))) . bq) * (- (bq * <i>))) * (((bq + 1) + (0 * <i>)) ") is complex set
((Partial_Sums ((bq * <i>))) *') . 0 is complex Element of COMPLEX
(Partial_Sums ((bq * <i>))) . 0 is complex Element of COMPLEX
((Partial_Sums ((bq * <i>))) . 0) *' is complex Element of COMPLEX
Re ((Partial_Sums ((bq * <i>))) . 0) is complex real ext-real Element of REAL
Im ((Partial_Sums ((bq * <i>))) . 0) is complex real ext-real Element of REAL
(Im ((Partial_Sums ((bq * <i>))) . 0)) * <i> is complex set
(Re ((Partial_Sums ((bq * <i>))) . 0)) - ((Im ((Partial_Sums ((bq * <i>))) . 0)) * <i>) is complex set
- ((Im ((Partial_Sums ((bq * <i>))) . 0)) * <i>) is complex set
(Re ((Partial_Sums ((bq * <i>))) . 0)) + (- ((Im ((Partial_Sums ((bq * <i>))) . 0)) * <i>)) is complex set
((bq * <i>)) . 0 is complex Element of COMPLEX
(((bq * <i>)) . 0) *' is complex Element of COMPLEX
Re (((bq * <i>)) . 0) is complex real ext-real Element of REAL
Im (((bq * <i>)) . 0) is complex real ext-real Element of REAL
(Im (((bq * <i>)) . 0)) * <i> is complex set
(Re (((bq * <i>)) . 0)) - ((Im (((bq * <i>)) . 0)) * <i>) is complex set
- ((Im (((bq * <i>)) . 0)) * <i>) is complex set
(Re (((bq * <i>)) . 0)) + (- ((Im (((bq * <i>)) . 0)) * <i>)) is complex set
((- (bq * <i>))) . 0 is complex Element of COMPLEX
(Partial_Sums ((- (bq * <i>)))) . 0 is complex Element of COMPLEX
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums ((bq * <i>))) *') . bq is complex Element of COMPLEX
(Partial_Sums ((- (bq * <i>)))) . bq is complex Element of COMPLEX
bq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums ((bq * <i>))) *') . (bq + 1) is complex Element of COMPLEX
(Partial_Sums ((- (bq * <i>)))) . (bq + 1) is complex Element of COMPLEX
(Partial_Sums ((bq * <i>))) . (bq + 1) is complex Element of COMPLEX
((Partial_Sums ((bq * <i>))) . (bq + 1)) *' is complex Element of COMPLEX
Re ((Partial_Sums ((bq * <i>))) . (bq + 1)) is complex real ext-real Element of REAL
Im ((Partial_Sums ((bq * <i>))) . (bq + 1)) is complex real ext-real Element of REAL
(Im ((Partial_Sums ((bq * <i>))) . (bq + 1))) * <i> is complex set
(Re ((Partial_Sums ((bq * <i>))) . (bq + 1))) - ((Im ((Partial_Sums ((bq * <i>))) . (bq + 1))) * <i>) is complex set
- ((Im ((Partial_Sums ((bq * <i>))) . (bq + 1))) * <i>) is complex set
(Re ((Partial_Sums ((bq * <i>))) . (bq + 1))) + (- ((Im ((Partial_Sums ((bq * <i>))) . (bq + 1))) * <i>)) is complex set
(Partial_Sums ((bq * <i>))) . bq is complex Element of COMPLEX
((bq * <i>)) . (bq + 1) is complex Element of COMPLEX
((Partial_Sums ((bq * <i>))) . bq) + (((bq * <i>)) . (bq + 1)) is complex Element of COMPLEX
(((Partial_Sums ((bq * <i>))) . bq) + (((bq * <i>)) . (bq + 1))) *' is complex Element of COMPLEX
Re (((Partial_Sums ((bq * <i>))) . bq) + (((bq * <i>)) . (bq + 1))) is complex real ext-real Element of REAL
Im (((Partial_Sums ((bq * <i>))) . bq) + (((bq * <i>)) . (bq + 1))) is complex real ext-real Element of REAL
(Im (((Partial_Sums ((bq * <i>))) . bq) + (((bq * <i>)) . (bq + 1)))) * <i> is complex set
(Re (((Partial_Sums ((bq * <i>))) . bq) + (((bq * <i>)) . (bq + 1)))) - ((Im (((Partial_Sums ((bq * <i>))) . bq) + (((bq * <i>)) . (bq + 1)))) * <i>) is complex set
- ((Im (((Partial_Sums ((bq * <i>))) . bq) + (((bq * <i>)) . (bq + 1)))) * <i>) is complex set
(Re (((Partial_Sums ((bq * <i>))) . bq) + (((bq * <i>)) . (bq + 1)))) + (- ((Im (((Partial_Sums ((bq * <i>))) . bq) + (((bq * <i>)) . (bq + 1)))) * <i>)) is complex set
((Partial_Sums ((bq * <i>))) . bq) *' is complex Element of COMPLEX
Re ((Partial_Sums ((bq * <i>))) . bq) is complex real ext-real Element of REAL
Im ((Partial_Sums ((bq * <i>))) . bq) is complex real ext-real Element of REAL
(Im ((Partial_Sums ((bq * <i>))) . bq)) * <i> is complex set
(Re ((Partial_Sums ((bq * <i>))) . bq)) - ((Im ((Partial_Sums ((bq * <i>))) . bq)) * <i>) is complex set
- ((Im ((Partial_Sums ((bq * <i>))) . bq)) * <i>) is complex set
(Re ((Partial_Sums ((bq * <i>))) . bq)) + (- ((Im ((Partial_Sums ((bq * <i>))) . bq)) * <i>)) is complex set
(((bq * <i>)) . (bq + 1)) *' is complex Element of COMPLEX
Re (((bq * <i>)) . (bq + 1)) is complex real ext-real Element of REAL
Im (((bq * <i>)) . (bq + 1)) is complex real ext-real Element of REAL
(Im (((bq * <i>)) . (bq + 1))) * <i> is complex set
(Re (((bq * <i>)) . (bq + 1))) - ((Im (((bq * <i>)) . (bq + 1))) * <i>) is complex set
- ((Im (((bq * <i>)) . (bq + 1))) * <i>) is complex set
(Re (((bq * <i>)) . (bq + 1))) + (- ((Im (((bq * <i>)) . (bq + 1))) * <i>)) is complex set
(((Partial_Sums ((bq * <i>))) . bq) *') + ((((bq * <i>)) . (bq + 1)) *') is complex Element of COMPLEX
((Partial_Sums ((- (bq * <i>)))) . bq) + ((((bq * <i>)) . (bq + 1)) *') is complex Element of COMPLEX
((- (bq * <i>))) . (bq + 1) is complex Element of COMPLEX
((Partial_Sums ((- (bq * <i>)))) . bq) + (((- (bq * <i>))) . (bq + 1)) is complex Element of COMPLEX
bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is complex Element of COMPLEX
() . (bq * <i>) is complex set
((bq * <i>)) *' is complex Element of COMPLEX
Re ((bq * <i>)) is complex real ext-real Element of REAL
Im ((bq * <i>)) is complex real ext-real Element of REAL
(Im ((bq * <i>))) * <i> is complex set
(Re ((bq * <i>))) - ((Im ((bq * <i>))) * <i>) is complex set
- ((Im ((bq * <i>))) * <i>) is complex set
(Re ((bq * <i>))) + (- ((Im ((bq * <i>))) * <i>)) is complex set
- (bq * <i>) is complex set
((- (bq * <i>))) is complex Element of COMPLEX
() . (- (bq * <i>)) is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
(Sum ((bq * <i>))) *' is complex Element of COMPLEX
Re (Sum ((bq * <i>))) is complex real ext-real Element of REAL
Im (Sum ((bq * <i>))) is complex real ext-real Element of REAL
(Im (Sum ((bq * <i>)))) * <i> is complex set
(Re (Sum ((bq * <i>)))) - ((Im (Sum ((bq * <i>)))) * <i>) is complex set
- ((Im (Sum ((bq * <i>)))) * <i>) is complex set
(Re (Sum ((bq * <i>)))) + (- ((Im (Sum ((bq * <i>)))) * <i>)) is complex set
((- (bq * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((- (bq * <i>))) is complex Element of COMPLEX
Partial_Sums ((- (bq * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((- (bq * <i>)))) is complex Element of COMPLEX
bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
|.(Sum ((bq * <i>))).| is complex real ext-real Element of REAL
Re (Sum ((bq * <i>))) is complex real ext-real Element of REAL
(Re (Sum ((bq * <i>)))) ^2 is complex real ext-real Element of REAL
(Re (Sum ((bq * <i>)))) * (Re (Sum ((bq * <i>)))) is complex real ext-real set
Im (Sum ((bq * <i>))) is complex real ext-real Element of REAL
(Im (Sum ((bq * <i>)))) ^2 is complex real ext-real Element of REAL
(Im (Sum ((bq * <i>)))) * (Im (Sum ((bq * <i>)))) is complex real ext-real set
((Re (Sum ((bq * <i>)))) ^2) + ((Im (Sum ((bq * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Sum ((bq * <i>)))) ^2) + ((Im (Sum ((bq * <i>)))) ^2)) is complex real ext-real Element of REAL
bq is complex real ext-real set
() . bq is complex real ext-real Element of REAL
abs (() . bq) is complex real ext-real Element of REAL
Re (() . bq) is complex real ext-real Element of REAL
(Re (() . bq)) ^2 is complex real ext-real Element of REAL
(Re (() . bq)) * (Re (() . bq)) is complex real ext-real set
Im (() . bq) is complex real ext-real Element of REAL
(Im (() . bq)) ^2 is complex real ext-real Element of REAL
(Im (() . bq)) * (Im (() . bq)) is complex real ext-real set
((Re (() . bq)) ^2) + ((Im (() . bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (() . bq)) ^2) + ((Im (() . bq)) ^2)) is complex real ext-real Element of REAL
() . bq is complex real ext-real Element of REAL
abs (() . bq) is complex real ext-real Element of REAL
Re (() . bq) is complex real ext-real Element of REAL
(Re (() . bq)) ^2 is complex real ext-real Element of REAL
(Re (() . bq)) * (Re (() . bq)) is complex real ext-real set
Im (() . bq) is complex real ext-real Element of REAL
(Im (() . bq)) ^2 is complex real ext-real Element of REAL
(Im (() . bq)) * (Im (() . bq)) is complex real ext-real set
((Re (() . bq)) ^2) + ((Im (() . bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (() . bq)) ^2) + ((Im (() . bq)) ^2)) is complex real ext-real Element of REAL
- (bq * <i>) is complex set
((- (bq * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((- (bq * <i>))) is complex Element of COMPLEX
Partial_Sums ((- (bq * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((- (bq * <i>)))) is complex Element of COMPLEX
(Sum ((bq * <i>))) * (Sum ((- (bq * <i>)))) is complex Element of COMPLEX
(bq * <i>) + (- (bq * <i>)) is complex set
(((bq * <i>) + (- (bq * <i>)))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (((bq * <i>) + (- (bq * <i>)))) is complex Element of COMPLEX
Partial_Sums (((bq * <i>) + (- (bq * <i>)))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((bq * <i>) + (- (bq * <i>))))) is complex Element of COMPLEX
|.(Sum ((bq * <i>))).| * |.(Sum ((bq * <i>))).| is complex real ext-real Element of REAL
(Sum ((bq * <i>))) * (Sum ((bq * <i>))) is complex Element of COMPLEX
|.((Sum ((bq * <i>))) * (Sum ((bq * <i>)))).| is complex real ext-real Element of REAL
Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
(Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2 is complex real ext-real Element of REAL
(Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) * (Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) is complex real ext-real set
Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
(Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2 is complex real ext-real Element of REAL
(Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) * (Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) is complex real ext-real set
((Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2) + ((Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2) + ((Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2)) is complex real ext-real Element of REAL
(Sum ((bq * <i>))) *' is complex Element of COMPLEX
(Im (Sum ((bq * <i>)))) * <i> is complex set
(Re (Sum ((bq * <i>)))) - ((Im (Sum ((bq * <i>)))) * <i>) is complex set
- ((Im (Sum ((bq * <i>)))) * <i>) is complex set
(Re (Sum ((bq * <i>)))) + (- ((Im (Sum ((bq * <i>)))) * <i>)) is complex set
(Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *') is complex Element of COMPLEX
|.((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *')).| is complex real ext-real Element of REAL
Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *')) is complex real ext-real Element of REAL
(Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2 is complex real ext-real Element of REAL
(Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) * (Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) is complex real ext-real set
Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *')) is complex real ext-real Element of REAL
(Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2 is complex real ext-real Element of REAL
(Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) * (Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) is complex real ext-real set
((Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2) + ((Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2) + ((Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2)) is complex real ext-real Element of REAL
1 / |.(Sum ((bq * <i>))).| is complex real ext-real Element of REAL
|.(Sum ((bq * <i>))).| " is complex real ext-real set
1 * (|.(Sum ((bq * <i>))).| ") is complex real ext-real set
- 1 is non empty complex real ext-real non positive negative Element of REAL
() . bq is complex real ext-real Element of REAL
abs (() . bq) is complex real ext-real Element of REAL
Re (() . bq) is complex real ext-real Element of REAL
(Re (() . bq)) ^2 is complex real ext-real Element of REAL
(Re (() . bq)) * (Re (() . bq)) is complex real ext-real set
Im (() . bq) is complex real ext-real Element of REAL
(Im (() . bq)) ^2 is complex real ext-real Element of REAL
(Im (() . bq)) * (Im (() . bq)) is complex real ext-real set
((Re (() . bq)) ^2) + ((Im (() . bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (() . bq)) ^2) + ((Im (() . bq)) ^2)) is complex real ext-real Element of REAL
abs (Im (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
Re (Im (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
(Re (Im (Sum ((bq * <i>))))) ^2 is complex real ext-real Element of REAL
(Re (Im (Sum ((bq * <i>))))) * (Re (Im (Sum ((bq * <i>))))) is complex real ext-real set
Im (Im (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
(Im (Im (Sum ((bq * <i>))))) ^2 is complex real ext-real Element of REAL
(Im (Im (Sum ((bq * <i>))))) * (Im (Im (Sum ((bq * <i>))))) is complex real ext-real set
((Re (Im (Sum ((bq * <i>))))) ^2) + ((Im (Im (Sum ((bq * <i>))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Im (Sum ((bq * <i>))))) ^2) + ((Im (Im (Sum ((bq * <i>))))) ^2)) is complex real ext-real Element of REAL
() . bq is complex real ext-real Element of REAL
abs (() . bq) is complex real ext-real Element of REAL
Re (() . bq) is complex real ext-real Element of REAL
(Re (() . bq)) ^2 is complex real ext-real Element of REAL
(Re (() . bq)) * (Re (() . bq)) is complex real ext-real set
Im (() . bq) is complex real ext-real Element of REAL
(Im (() . bq)) ^2 is complex real ext-real Element of REAL
(Im (() . bq)) * (Im (() . bq)) is complex real ext-real set
((Re (() . bq)) ^2) + ((Im (() . bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (() . bq)) ^2) + ((Im (() . bq)) ^2)) is complex real ext-real Element of REAL
abs (Re (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
Re (Re (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
(Re (Re (Sum ((bq * <i>))))) ^2 is complex real ext-real Element of REAL
(Re (Re (Sum ((bq * <i>))))) * (Re (Re (Sum ((bq * <i>))))) is complex real ext-real set
Im (Re (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
(Im (Re (Sum ((bq * <i>))))) ^2 is complex real ext-real Element of REAL
(Im (Re (Sum ((bq * <i>))))) * (Im (Re (Sum ((bq * <i>))))) is complex real ext-real set
((Re (Re (Sum ((bq * <i>))))) ^2) + ((Im (Re (Sum ((bq * <i>))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Re (Sum ((bq * <i>))))) ^2) + ((Im (Re (Sum ((bq * <i>))))) ^2)) is complex real ext-real Element of REAL
bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is complex Element of COMPLEX
() . (bq * <i>) is complex set
|.((bq * <i>)).| is complex real ext-real Element of REAL
Re ((bq * <i>)) is complex real ext-real Element of REAL
(Re ((bq * <i>))) ^2 is complex real ext-real Element of REAL
(Re ((bq * <i>))) * (Re ((bq * <i>))) is complex real ext-real set
Im ((bq * <i>)) is complex real ext-real Element of REAL
(Im ((bq * <i>))) ^2 is complex real ext-real Element of REAL
(Im ((bq * <i>))) * (Im ((bq * <i>))) is complex real ext-real set
((Re ((bq * <i>))) ^2) + ((Im ((bq * <i>))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((bq * <i>))) ^2) + ((Im ((bq * <i>))) ^2)) is complex real ext-real Element of REAL
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
|.(Sum ((bq * <i>))).| is complex real ext-real Element of REAL
Re (Sum ((bq * <i>))) is complex real ext-real Element of REAL
(Re (Sum ((bq * <i>)))) ^2 is complex real ext-real Element of REAL
(Re (Sum ((bq * <i>)))) * (Re (Sum ((bq * <i>)))) is complex real ext-real set
Im (Sum ((bq * <i>))) is complex real ext-real Element of REAL
(Im (Sum ((bq * <i>)))) ^2 is complex real ext-real Element of REAL
(Im (Sum ((bq * <i>)))) * (Im (Sum ((bq * <i>)))) is complex real ext-real set
((Re (Sum ((bq * <i>)))) ^2) + ((Im (Sum ((bq * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Sum ((bq * <i>)))) ^2) + ((Im (Sum ((bq * <i>)))) ^2)) is complex real ext-real Element of REAL
bq is complex real ext-real set
(bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
abs (bq) is complex real ext-real Element of REAL
Re (bq) is complex real ext-real Element of REAL
(Re (bq)) ^2 is complex real ext-real Element of REAL
(Re (bq)) * (Re (bq)) is complex real ext-real set
Im (bq) is complex real ext-real Element of REAL
(Im (bq)) ^2 is complex real ext-real Element of REAL
(Im (bq)) * (Im (bq)) is complex real ext-real set
((Re (bq)) ^2) + ((Im (bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (bq)) ^2) + ((Im (bq)) ^2)) is complex real ext-real Element of REAL
(bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
abs (bq) is complex real ext-real Element of REAL
Re (bq) is complex real ext-real Element of REAL
(Re (bq)) ^2 is complex real ext-real Element of REAL
(Re (bq)) * (Re (bq)) is complex real ext-real set
Im (bq) is complex real ext-real Element of REAL
(Im (bq)) ^2 is complex real ext-real Element of REAL
(Im (bq)) * (Im (bq)) is complex real ext-real set
((Re (bq)) ^2) + ((Im (bq)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (bq)) ^2) + ((Im (bq)) ^2)) is complex real ext-real Element of REAL
bq is complex real ext-real set
() . bq is complex real ext-real Element of REAL
(() . bq) ^2 is complex real ext-real Element of REAL
(() . bq) * (() . bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
(() . bq) ^2 is complex real ext-real Element of REAL
(() . bq) * (() . bq) is complex real ext-real set
((() . bq) ^2) + ((() . bq) ^2) is complex real ext-real Element of REAL
(() . bq) * (() . bq) is complex real ext-real Element of REAL
(() . bq) * (() . bq) is complex real ext-real Element of REAL
((() . bq) * (() . bq)) + ((() . bq) * (() . bq)) is complex real ext-real Element of REAL
bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
- (bq * <i>) is complex set
((- (bq * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((- (bq * <i>))) is complex Element of COMPLEX
Partial_Sums ((- (bq * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((- (bq * <i>)))) is complex Element of COMPLEX
(Sum ((bq * <i>))) * (Sum ((- (bq * <i>)))) is complex Element of COMPLEX
(bq * <i>) + (- (bq * <i>)) is complex set
(((bq * <i>) + (- (bq * <i>)))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (((bq * <i>) + (- (bq * <i>)))) is complex Element of COMPLEX
Partial_Sums (((bq * <i>) + (- (bq * <i>)))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((bq * <i>) + (- (bq * <i>))))) is complex Element of COMPLEX
Re (Sum ((bq * <i>))) is complex real ext-real Element of REAL
(Re (Sum ((bq * <i>)))) ^2 is complex real ext-real Element of REAL
(Re (Sum ((bq * <i>)))) * (Re (Sum ((bq * <i>)))) is complex real ext-real set
((Re (Sum ((bq * <i>)))) ^2) + ((() . bq) ^2) is complex real ext-real Element of REAL
Im (Sum ((bq * <i>))) is complex real ext-real Element of REAL
(Im (Sum ((bq * <i>)))) ^2 is complex real ext-real Element of REAL
(Im (Sum ((bq * <i>)))) * (Im (Sum ((bq * <i>)))) is complex real ext-real set
((Re (Sum ((bq * <i>)))) ^2) + ((Im (Sum ((bq * <i>)))) ^2) is complex real ext-real Element of REAL
(Sum ((bq * <i>))) * (Sum ((bq * <i>))) is complex Element of COMPLEX
|.((Sum ((bq * <i>))) * (Sum ((bq * <i>)))).| is complex real ext-real Element of REAL
Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
(Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2 is complex real ext-real Element of REAL
(Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) * (Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) is complex real ext-real set
Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
(Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2 is complex real ext-real Element of REAL
(Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) * (Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) is complex real ext-real set
((Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2) + ((Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2) + ((Im ((Sum ((bq * <i>))) * (Sum ((bq * <i>))))) ^2)) is complex real ext-real Element of REAL
(Sum ((bq * <i>))) *' is complex Element of COMPLEX
(Im (Sum ((bq * <i>)))) * <i> is complex set
(Re (Sum ((bq * <i>)))) - ((Im (Sum ((bq * <i>)))) * <i>) is complex set
- ((Im (Sum ((bq * <i>)))) * <i>) is complex set
(Re (Sum ((bq * <i>)))) + (- ((Im (Sum ((bq * <i>)))) * <i>)) is complex set
(Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *') is complex Element of COMPLEX
|.((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *')).| is complex real ext-real Element of REAL
Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *')) is complex real ext-real Element of REAL
(Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2 is complex real ext-real Element of REAL
(Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) * (Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) is complex real ext-real set
Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *')) is complex real ext-real Element of REAL
(Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2 is complex real ext-real Element of REAL
(Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) * (Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) is complex real ext-real set
((Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2) + ((Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2) + ((Im ((Sum ((bq * <i>))) * ((Sum ((bq * <i>))) *'))) ^2)) is complex real ext-real Element of REAL
bq is complex real ext-real set
(bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
(bq) ^2 is complex real ext-real set
(bq) * (bq) is complex real ext-real set
(bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
(bq) ^2 is complex real ext-real set
(bq) * (bq) is complex real ext-real set
((bq) ^2) + ((bq) ^2) is complex real ext-real set
bq is complex real ext-real set
(bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
(bq) * (bq) is complex real ext-real set
(bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
(bq) * (bq) is complex real ext-real set
((bq) * (bq)) + ((bq) * (bq)) is complex real ext-real set
() . 0 is complex real ext-real Element of REAL
() . 0 is complex real ext-real Element of REAL
bq is complex real ext-real set
- bq is complex real ext-real set
() . (- bq) is complex real ext-real Element of REAL
() . bq is complex real ext-real Element of REAL
() . (- bq) is complex real ext-real Element of REAL
() . bq is complex real ext-real Element of REAL
- (() . bq) is complex real ext-real Element of REAL
((0 * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((0 * <i>)) is complex Element of COMPLEX
Partial_Sums ((0 * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((0 * <i>))) is complex Element of COMPLEX
Re (Sum ((0 * <i>))) is complex real ext-real Element of REAL
Im (Sum ((0 * <i>))) is complex real ext-real Element of REAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of REAL
bq is complex real ext-real Element of REAL
- bq is complex real ext-real Element of REAL
(- bq) * <i> is complex set
(- 0) + ((- bq) * <i>) is complex set
(((- 0) + ((- bq) * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (((- 0) + ((- bq) * <i>))) is complex Element of COMPLEX
Partial_Sums (((- 0) + ((- bq) * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((- 0) + ((- bq) * <i>)))) is complex Element of COMPLEX
Re (Sum (((- 0) + ((- bq) * <i>)))) is complex real ext-real Element of REAL
bq * <i> is complex set
- (bq * <i>) is complex set
((- (bq * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((- (bq * <i>))) is complex Element of COMPLEX
Partial_Sums ((- (bq * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((- (bq * <i>)))) is complex Element of COMPLEX
Re (Sum ((- (bq * <i>)))) is complex real ext-real Element of REAL
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
(Sum ((bq * <i>))) *' is complex Element of COMPLEX
Re (Sum ((bq * <i>))) is complex real ext-real Element of REAL
Im (Sum ((bq * <i>))) is complex real ext-real Element of REAL
(Im (Sum ((bq * <i>)))) * <i> is complex set
(Re (Sum ((bq * <i>)))) - ((Im (Sum ((bq * <i>)))) * <i>) is complex set
- ((Im (Sum ((bq * <i>)))) * <i>) is complex set
(Re (Sum ((bq * <i>)))) + (- ((Im (Sum ((bq * <i>)))) * <i>)) is complex set
Re ((Sum ((bq * <i>))) *') is complex real ext-real Element of REAL
(() . bq) * <i> is complex set
(() . bq) + ((() . bq) * <i>) is complex set
((() . bq) + ((() . bq) * <i>)) *' is complex Element of COMPLEX
Re ((() . bq) + ((() . bq) * <i>)) is complex real ext-real Element of REAL
Im ((() . bq) + ((() . bq) * <i>)) is complex real ext-real Element of REAL
(Im ((() . bq) + ((() . bq) * <i>))) * <i> is complex set
(Re ((() . bq) + ((() . bq) * <i>))) - ((Im ((() . bq) + ((() . bq) * <i>))) * <i>) is complex set
- ((Im ((() . bq) + ((() . bq) * <i>))) * <i>) is complex set
(Re ((() . bq) + ((() . bq) * <i>))) + (- ((Im ((() . bq) + ((() . bq) * <i>))) * <i>)) is complex set
Re (((() . bq) + ((() . bq) * <i>)) *') is complex real ext-real Element of REAL
(- (() . bq)) * <i> is complex set
(() . bq) + ((- (() . bq)) * <i>) is complex set
Re ((() . bq) + ((- (() . bq)) * <i>)) is complex real ext-real Element of REAL
Im (Sum (((- 0) + ((- bq) * <i>)))) is complex real ext-real Element of REAL
Im (Sum ((- (bq * <i>)))) is complex real ext-real Element of REAL
Im ((Sum ((bq * <i>))) *') is complex real ext-real Element of REAL
Im (((() . bq) + ((() . bq) * <i>)) *') is complex real ext-real Element of REAL
Im ((() . bq) + ((- (() . bq)) * <i>)) is complex real ext-real Element of REAL
(0) is complex real ext-real Element of REAL
(0) is complex real ext-real Element of REAL
bq is complex real ext-real set
- bq is complex real ext-real set
((- bq)) is complex real ext-real set
() . (- bq) is complex real ext-real Element of REAL
(bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
bq is complex real ext-real set
- bq is complex real ext-real set
((- bq)) is complex real ext-real set
() . (- bq) is complex real ext-real Element of REAL
(bq) is complex real ext-real set
() . bq is complex real ext-real Element of REAL
- (bq) is complex real ext-real set
- 1 is non empty complex real ext-real non positive negative Element of REAL
bq is complex real ext-real set
bq is complex set
bq |^ 2 is complex set
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . 2 is complex Element of COMPLEX
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
2 * bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
bq |^ (2 * bq) is complex set
(bq GeoSeq) . (2 * bq) is complex Element of COMPLEX
bq |^ bq is complex set
(bq GeoSeq) . bq is complex Element of COMPLEX
(bq |^ bq) |^ 2 is complex set
(bq |^ bq) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq |^ bq) GeoSeq) . 2 is complex Element of COMPLEX
(bq |^ 2) |^ bq is complex set
(bq |^ 2) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq |^ 2) GeoSeq) . bq is complex Element of COMPLEX
2 * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() V62() V63() V98() Element of NAT
bq |^ (2 * 0) is complex set
(bq GeoSeq) . (2 * 0) is complex Element of COMPLEX
1 * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
1 |^ 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
1 GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(1 GeoSeq) . 1 is complex Element of COMPLEX
(1 |^ 1) * 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
1 |^ (1 + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 GeoSeq) . (1 + 1) is complex Element of COMPLEX
bq |^ 0 is complex set
(bq GeoSeq) . 0 is complex Element of COMPLEX
(bq |^ 0) |^ 2 is complex set
(bq |^ 0) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq |^ 0) GeoSeq) . 2 is complex Element of COMPLEX
(bq |^ 2) |^ 0 is complex set
((bq |^ 2) GeoSeq) . 0 is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
2 * a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
bq |^ (2 * a) is complex set
(bq GeoSeq) . (2 * a) is complex Element of COMPLEX
bq |^ a is complex set
(bq GeoSeq) . a is complex Element of COMPLEX
(bq |^ a) |^ 2 is complex set
(bq |^ a) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq |^ a) GeoSeq) . 2 is complex Element of COMPLEX
(bq |^ 2) |^ a is complex set
((bq |^ 2) GeoSeq) . a is complex Element of COMPLEX
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
2 * (a + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
bq |^ (2 * (a + 1)) is complex set
(bq GeoSeq) . (2 * (a + 1)) is complex Element of COMPLEX
bq |^ (a + 1) is complex set
(bq GeoSeq) . (a + 1) is complex Element of COMPLEX
(bq |^ (a + 1)) |^ 2 is complex set
(bq |^ (a + 1)) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq |^ (a + 1)) GeoSeq) . 2 is complex Element of COMPLEX
(bq |^ 2) |^ (a + 1) is complex set
((bq |^ 2) GeoSeq) . (a + 1) is complex Element of COMPLEX
(2 * a) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
((2 * a) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(bq GeoSeq) . (((2 * a) + 1) + 1) is complex Element of COMPLEX
(bq GeoSeq) . ((2 * a) + 1) is complex Element of COMPLEX
((bq GeoSeq) . ((2 * a) + 1)) * bq is complex set
(bq |^ (2 * a)) * bq is complex set
((bq |^ (2 * a)) * bq) * bq is complex set
((bq |^ a) GeoSeq) . (1 + 1) is complex Element of COMPLEX
(((bq |^ a) GeoSeq) . (1 + 1)) * bq is complex set
((((bq |^ a) GeoSeq) . (1 + 1)) * bq) * bq is complex set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((bq |^ a) GeoSeq) . (0 + 1) is complex Element of COMPLEX
(((bq |^ a) GeoSeq) . (0 + 1)) * (bq |^ a) is complex set
((((bq |^ a) GeoSeq) . (0 + 1)) * (bq |^ a)) * bq is complex set
(((((bq |^ a) GeoSeq) . (0 + 1)) * (bq |^ a)) * bq) * bq is complex set
((bq |^ a) GeoSeq) . 0 is complex Element of COMPLEX
(((bq |^ a) GeoSeq) . 0) * (bq |^ a) is complex set
((((bq |^ a) GeoSeq) . 0) * (bq |^ a)) * (bq |^ a) is complex set
(((((bq |^ a) GeoSeq) . 0) * (bq |^ a)) * (bq |^ a)) * bq is complex set
((((((bq |^ a) GeoSeq) . 0) * (bq |^ a)) * (bq |^ a)) * bq) * bq is complex set
1r * (bq |^ a) is complex set
(1r * (bq |^ a)) * (bq |^ a) is complex set
((1r * (bq |^ a)) * (bq |^ a)) * bq is complex set
(((1r * (bq |^ a)) * (bq |^ a)) * bq) * bq is complex set
((bq GeoSeq) . a) * bq is complex set
1r * (((bq GeoSeq) . a) * bq) is complex set
(bq |^ a) * bq is complex set
(1r * (((bq GeoSeq) . a) * bq)) * ((bq |^ a) * bq) is complex set
1r * ((bq GeoSeq) . (a + 1)) is complex Element of COMPLEX
(1r * ((bq GeoSeq) . (a + 1))) * (((bq GeoSeq) . a) * bq) is complex set
1r * (bq |^ (a + 1)) is complex set
(1r * (bq |^ (a + 1))) * ((bq GeoSeq) . (a + 1)) is complex set
((bq |^ (a + 1)) GeoSeq) . 0 is complex Element of COMPLEX
(((bq |^ (a + 1)) GeoSeq) . 0) * (bq |^ (a + 1)) is complex set
((((bq |^ (a + 1)) GeoSeq) . 0) * (bq |^ (a + 1))) * (bq |^ (a + 1)) is complex set
((bq |^ (a + 1)) GeoSeq) . (0 + 1) is complex Element of COMPLEX
(((bq |^ (a + 1)) GeoSeq) . (0 + 1)) * (bq |^ (a + 1)) is complex set
(0 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((bq |^ (a + 1)) GeoSeq) . ((0 + 1) + 1) is complex Element of COMPLEX
1r * bq is complex set
(1r * bq) * bq is complex set
((bq |^ 2) |^ a) * ((1r * bq) * bq) is complex set
((bq GeoSeq) . 0) * bq is complex set
(((bq GeoSeq) . 0) * bq) * bq is complex set
((bq |^ 2) |^ a) * ((((bq GeoSeq) . 0) * bq) * bq) is complex set
(bq GeoSeq) . (0 + 1) is complex Element of COMPLEX
((bq GeoSeq) . (0 + 1)) * bq is complex set
((bq |^ 2) |^ a) * (((bq GeoSeq) . (0 + 1)) * bq) is complex set
(bq GeoSeq) . ((0 + 1) + 1) is complex Element of COMPLEX
((bq |^ 2) |^ a) * ((bq GeoSeq) . ((0 + 1) + 1)) is complex set
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
2 * bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(- 1) |^ bq is complex real ext-real Element of REAL
(- 1) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((- 1) GeoSeq) . bq is complex Element of COMPLEX
(2 * bq) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
bq is complex real ext-real Element of REAL
bq * <i> is complex set
(bq * <i>) |^ (2 * bq) is complex set
(bq * <i>) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq * <i>) GeoSeq) . (2 * bq) is complex Element of COMPLEX
bq |^ (2 * bq) is complex real ext-real Element of REAL
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . (2 * bq) is complex Element of COMPLEX
((- 1) |^ bq) * (bq |^ (2 * bq)) is complex real ext-real Element of REAL
(bq * <i>) |^ ((2 * bq) + 1) is complex set
((bq * <i>) GeoSeq) . ((2 * bq) + 1) is complex Element of COMPLEX
bq |^ ((2 * bq) + 1) is complex real ext-real Element of REAL
(bq GeoSeq) . ((2 * bq) + 1) is complex Element of COMPLEX
((- 1) |^ bq) * (bq |^ ((2 * bq) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ bq) * (bq |^ ((2 * bq) + 1))) * <i> is complex set
(- 1) |^ 0 is complex real ext-real Element of REAL
((- 1) GeoSeq) . 0 is complex Element of COMPLEX
2 * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() V62() V63() V98() Element of NAT
bq |^ (2 * 0) is complex real ext-real Element of REAL
(bq GeoSeq) . (2 * 0) is complex Element of COMPLEX
((- 1) |^ 0) * (bq |^ (2 * 0)) is complex real ext-real Element of REAL
(((- 1) |^ 0) * (bq |^ (2 * 0))) * <i> is complex set
bq GeoSeq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(bq GeoSeq) . 0 is complex real ext-real Element of REAL
((- 1) |^ 0) * ((bq GeoSeq) . 0) is complex real ext-real Element of REAL
(((- 1) |^ 0) * ((bq GeoSeq) . 0)) * <i> is complex set
((- 1) |^ 0) * 1 is complex real ext-real Element of REAL
(((- 1) |^ 0) * 1) * <i> is complex set
(- 1) GeoSeq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((- 1) GeoSeq) . 0 is complex real ext-real Element of REAL
(((- 1) GeoSeq) . 0) * <i> is complex set
1 * <i> is complex set
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(bq * <i>) |^ ((2 * 0) + 1) is complex set
((bq * <i>) GeoSeq) . ((2 * 0) + 1) is complex Element of COMPLEX
((bq * <i>) GeoSeq) . 0 is complex Element of COMPLEX
(((bq * <i>) GeoSeq) . 0) * (bq * <i>) is complex set
1 * (bq * <i>) is complex set
bq |^ ((2 * 0) + 1) is complex real ext-real Element of REAL
(bq GeoSeq) . ((2 * 0) + 1) is complex Element of COMPLEX
((- 1) |^ 0) * (bq |^ ((2 * 0) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ 0) * (bq |^ ((2 * 0) + 1))) * <i> is complex set
(bq GeoSeq) . ((2 * 0) + 1) is complex real ext-real Element of REAL
((- 1) |^ 0) * ((bq GeoSeq) . ((2 * 0) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ 0) * ((bq GeoSeq) . ((2 * 0) + 1))) * <i> is complex set
((bq GeoSeq) . 0) * bq is complex real ext-real Element of REAL
((- 1) |^ 0) * (((bq GeoSeq) . 0) * bq) is complex real ext-real Element of REAL
(((- 1) |^ 0) * (((bq GeoSeq) . 0) * bq)) * <i> is complex set
1 * bq is complex real ext-real Element of REAL
((- 1) |^ 0) * (1 * bq) is complex real ext-real Element of REAL
(((- 1) |^ 0) * (1 * bq)) * <i> is complex set
(((- 1) GeoSeq) . 0) * bq is complex real ext-real Element of REAL
((((- 1) GeoSeq) . 0) * bq) * <i> is complex set
(1 * bq) * <i> is complex set
(bq * <i>) |^ (2 * 0) is complex set
((bq * <i>) GeoSeq) . (2 * 0) is complex Element of COMPLEX
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
2 * a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(bq * <i>) |^ (2 * a) is complex set
((bq * <i>) GeoSeq) . (2 * a) is complex Element of COMPLEX
(- 1) |^ a is complex real ext-real Element of REAL
((- 1) GeoSeq) . a is complex Element of COMPLEX
bq |^ (2 * a) is complex real ext-real Element of REAL
(bq GeoSeq) . (2 * a) is complex Element of COMPLEX
((- 1) |^ a) * (bq |^ (2 * a)) is complex real ext-real Element of REAL
(2 * a) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(bq * <i>) |^ ((2 * a) + 1) is complex set
((bq * <i>) GeoSeq) . ((2 * a) + 1) is complex Element of COMPLEX
bq |^ ((2 * a) + 1) is complex real ext-real Element of REAL
(bq GeoSeq) . ((2 * a) + 1) is complex Element of COMPLEX
((- 1) |^ a) * (bq |^ ((2 * a) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ a) * (bq |^ ((2 * a) + 1))) * <i> is complex set
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
2 * (a + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(bq * <i>) |^ (2 * (a + 1)) is complex set
((bq * <i>) GeoSeq) . (2 * (a + 1)) is complex Element of COMPLEX
(- 1) |^ (a + 1) is complex real ext-real Element of REAL
((- 1) GeoSeq) . (a + 1) is complex Element of COMPLEX
bq |^ (2 * (a + 1)) is complex real ext-real Element of REAL
(bq GeoSeq) . (2 * (a + 1)) is complex Element of COMPLEX
((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1))) is complex real ext-real Element of REAL
(2 * (a + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(bq * <i>) |^ ((2 * (a + 1)) + 1) is complex set
((bq * <i>) GeoSeq) . ((2 * (a + 1)) + 1) is complex Element of COMPLEX
bq |^ ((2 * (a + 1)) + 1) is complex real ext-real Element of REAL
(bq GeoSeq) . ((2 * (a + 1)) + 1) is complex Element of COMPLEX
((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) * <i> is complex set
(bq * <i>) |^ 2 is complex set
((bq * <i>) GeoSeq) . 2 is complex Element of COMPLEX
((bq * <i>) |^ 2) |^ (a + 1) is complex set
((bq * <i>) |^ 2) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(((bq * <i>) |^ 2) GeoSeq) . (a + 1) is complex Element of COMPLEX
((bq * <i>) |^ 2) |^ a is complex set
(((bq * <i>) |^ 2) GeoSeq) . a is complex Element of COMPLEX
(((bq * <i>) |^ 2) |^ a) * ((bq * <i>) |^ 2) is complex set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((bq * <i>) GeoSeq) . (1 + 1) is complex Element of COMPLEX
(((- 1) |^ a) * (bq |^ (2 * a))) * (((bq * <i>) GeoSeq) . (1 + 1)) is complex set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((bq * <i>) GeoSeq) . (0 + 1) is complex Element of COMPLEX
(((bq * <i>) GeoSeq) . (0 + 1)) * (bq * <i>) is complex set
(((- 1) |^ a) * (bq |^ (2 * a))) * ((((bq * <i>) GeoSeq) . (0 + 1)) * (bq * <i>)) is complex set
((((bq * <i>) GeoSeq) . 0) * (bq * <i>)) * (bq * <i>) is complex set
(((- 1) |^ a) * (bq |^ (2 * a))) * (((((bq * <i>) GeoSeq) . 0) * (bq * <i>)) * (bq * <i>)) is complex set
1r * (bq * <i>) is complex set
(1r * (bq * <i>)) * (bq * <i>) is complex set
(((- 1) |^ a) * (bq |^ (2 * a))) * ((1r * (bq * <i>)) * (bq * <i>)) is complex set
((- 1) |^ a) * (- 1) is complex real ext-real Element of REAL
(((- 1) |^ a) * (- 1)) * (bq |^ (2 * a)) is complex real ext-real Element of REAL
((((- 1) |^ a) * (- 1)) * (bq |^ (2 * a))) * bq is complex real ext-real Element of REAL
(((((- 1) |^ a) * (- 1)) * (bq |^ (2 * a))) * bq) * bq is complex real ext-real Element of REAL
((- 1) GeoSeq) . a is complex real ext-real Element of REAL
(((- 1) GeoSeq) . a) * (- 1) is complex real ext-real Element of REAL
((((- 1) GeoSeq) . a) * (- 1)) * (bq |^ (2 * a)) is complex real ext-real Element of REAL
(((((- 1) GeoSeq) . a) * (- 1)) * (bq |^ (2 * a))) * bq is complex real ext-real Element of REAL
((((((- 1) GeoSeq) . a) * (- 1)) * (bq |^ (2 * a))) * bq) * bq is complex real ext-real Element of REAL
((- 1) GeoSeq) . (a + 1) is complex real ext-real Element of REAL
(((- 1) GeoSeq) . (a + 1)) * (bq |^ (2 * a)) is complex real ext-real Element of REAL
((((- 1) GeoSeq) . (a + 1)) * (bq |^ (2 * a))) * bq is complex real ext-real Element of REAL
(((((- 1) GeoSeq) . (a + 1)) * (bq |^ (2 * a))) * bq) * bq is complex real ext-real Element of REAL
((- 1) |^ (a + 1)) * (bq |^ (2 * a)) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * (bq |^ (2 * a))) * bq is complex real ext-real Element of REAL
((((- 1) |^ (a + 1)) * (bq |^ (2 * a))) * bq) * bq is complex real ext-real Element of REAL
(bq GeoSeq) . (2 * a) is complex real ext-real Element of REAL
((- 1) |^ (a + 1)) * ((bq GeoSeq) . (2 * a)) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * ((bq GeoSeq) . (2 * a))) * bq is complex real ext-real Element of REAL
((((- 1) |^ (a + 1)) * ((bq GeoSeq) . (2 * a))) * bq) * bq is complex real ext-real Element of REAL
((bq GeoSeq) . (2 * a)) * bq is complex real ext-real Element of REAL
((- 1) |^ (a + 1)) * (((bq GeoSeq) . (2 * a)) * bq) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * (((bq GeoSeq) . (2 * a)) * bq)) * bq is complex real ext-real Element of REAL
(bq GeoSeq) . ((2 * a) + 1) is complex real ext-real Element of REAL
((- 1) |^ (a + 1)) * ((bq GeoSeq) . ((2 * a) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * ((bq GeoSeq) . ((2 * a) + 1))) * bq is complex real ext-real Element of REAL
((bq GeoSeq) . ((2 * a) + 1)) * bq is complex real ext-real Element of REAL
((- 1) |^ (a + 1)) * (((bq GeoSeq) . ((2 * a) + 1)) * bq) is complex real ext-real Element of REAL
((2 * a) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(bq GeoSeq) . (((2 * a) + 1) + 1) is complex real ext-real Element of REAL
((- 1) |^ (a + 1)) * ((bq GeoSeq) . (((2 * a) + 1) + 1)) is complex real ext-real Element of REAL
((bq * <i>) GeoSeq) . (((2 * a) + 1) + 1) is complex Element of COMPLEX
(((bq * <i>) GeoSeq) . (((2 * a) + 1) + 1)) * (bq * <i>) is complex set
(((bq * <i>) GeoSeq) . ((2 * a) + 1)) * (bq * <i>) is complex set
((((bq * <i>) GeoSeq) . ((2 * a) + 1)) * (bq * <i>)) * (bq * <i>) is complex set
(((- 1) |^ a) * (- 1)) * (bq |^ ((2 * a) + 1)) is complex real ext-real Element of REAL
((((- 1) |^ a) * (- 1)) * (bq |^ ((2 * a) + 1))) * bq is complex real ext-real Element of REAL
(((((- 1) |^ a) * (- 1)) * (bq |^ ((2 * a) + 1))) * bq) * bq is complex real ext-real Element of REAL
((((((- 1) |^ a) * (- 1)) * (bq |^ ((2 * a) + 1))) * bq) * bq) * <i> is complex set
((((- 1) GeoSeq) . a) * (- 1)) * (bq |^ ((2 * a) + 1)) is complex real ext-real Element of REAL
(((((- 1) GeoSeq) . a) * (- 1)) * (bq |^ ((2 * a) + 1))) * bq is complex real ext-real Element of REAL
((((((- 1) GeoSeq) . a) * (- 1)) * (bq |^ ((2 * a) + 1))) * bq) * bq is complex real ext-real Element of REAL
(((((((- 1) GeoSeq) . a) * (- 1)) * (bq |^ ((2 * a) + 1))) * bq) * bq) * <i> is complex set
(((- 1) GeoSeq) . (a + 1)) * (bq |^ ((2 * a) + 1)) is complex real ext-real Element of REAL
((((- 1) GeoSeq) . (a + 1)) * (bq |^ ((2 * a) + 1))) * bq is complex real ext-real Element of REAL
(((((- 1) GeoSeq) . (a + 1)) * (bq |^ ((2 * a) + 1))) * bq) * bq is complex real ext-real Element of REAL
((((((- 1) GeoSeq) . (a + 1)) * (bq |^ ((2 * a) + 1))) * bq) * bq) * <i> is complex set
((- 1) |^ (a + 1)) * (bq |^ ((2 * a) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * (bq |^ ((2 * a) + 1))) * bq is complex real ext-real Element of REAL
((((- 1) |^ (a + 1)) * (bq |^ ((2 * a) + 1))) * bq) * bq is complex real ext-real Element of REAL
(((((- 1) |^ (a + 1)) * (bq |^ ((2 * a) + 1))) * bq) * bq) * <i> is complex set
((((- 1) |^ (a + 1)) * ((bq GeoSeq) . ((2 * a) + 1))) * bq) * bq is complex real ext-real Element of REAL
(((((- 1) |^ (a + 1)) * ((bq GeoSeq) . ((2 * a) + 1))) * bq) * bq) * <i> is complex set
(((- 1) |^ (a + 1)) * (((bq GeoSeq) . ((2 * a) + 1)) * bq)) * bq is complex real ext-real Element of REAL
((((- 1) |^ (a + 1)) * (((bq GeoSeq) . ((2 * a) + 1)) * bq)) * bq) * <i> is complex set
(((- 1) |^ (a + 1)) * ((bq GeoSeq) . (((2 * a) + 1) + 1))) * bq is complex real ext-real Element of REAL
((((- 1) |^ (a + 1)) * ((bq GeoSeq) . (((2 * a) + 1) + 1))) * bq) * <i> is complex set
((bq GeoSeq) . (((2 * a) + 1) + 1)) * bq is complex real ext-real Element of REAL
((- 1) |^ (a + 1)) * (((bq GeoSeq) . (((2 * a) + 1) + 1)) * bq) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * (((bq GeoSeq) . (((2 * a) + 1) + 1)) * bq)) * <i> is complex set
(bq GeoSeq) . ((2 * (a + 1)) + 1) is complex real ext-real Element of REAL
((- 1) |^ (a + 1)) * ((bq GeoSeq) . ((2 * (a + 1)) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * ((bq GeoSeq) . ((2 * (a + 1)) + 1))) * <i> is complex set
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . bq is complex real ext-real Element of REAL
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
2 * bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * bq) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
bq is complex real ext-real Element of REAL
(bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(Partial_Sums (bq)) . bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Im ((bq * <i>)) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
Partial_Sums (Im ((bq * <i>))) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(Partial_Sums (Im ((bq * <i>)))) . ((2 * bq) + 1) is complex real ext-real Element of REAL
(bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(Partial_Sums (bq)) . bq is complex real ext-real Element of REAL
Re ((bq * <i>)) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
Partial_Sums (Re ((bq * <i>))) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(Partial_Sums (Re ((bq * <i>)))) . (2 * bq) is complex real ext-real Element of REAL
(Partial_Sums (bq)) . 0 is complex real ext-real Element of REAL
(bq) . 0 is complex real ext-real Element of REAL
(- 1) |^ 0 is complex real ext-real Element of REAL
(- 1) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((- 1) GeoSeq) . 0 is complex Element of COMPLEX
2 * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() V62() V63() V98() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
bq |^ ((2 * 0) + 1) is complex real ext-real Element of REAL
bq GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(bq GeoSeq) . ((2 * 0) + 1) is complex Element of COMPLEX
((- 1) |^ 0) * (bq |^ ((2 * 0) + 1)) is complex real ext-real Element of REAL
((2 * 0) + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((2 * 0) + 1) is complex real ext-real Element of REAL
(((- 1) |^ 0) * (bq |^ ((2 * 0) + 1))) / (((2 * 0) + 1) !) is complex real ext-real Element of REAL
(((2 * 0) + 1) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ 0) * (bq |^ ((2 * 0) + 1))) * ((((2 * 0) + 1) !) ") is complex real ext-real set
bq GeoSeq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(bq GeoSeq) . (0 + 1) is complex real ext-real Element of REAL
(bq GeoSeq) . 0 is complex real ext-real Element of REAL
((bq GeoSeq) . 0) * bq is complex real ext-real Element of REAL
1 * bq is complex real ext-real Element of REAL
(0 !) * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(- 1) GeoSeq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((- 1) GeoSeq) . 0 is complex real ext-real Element of REAL
(Partial_Sums (bq)) . 0 is complex real ext-real Element of REAL
(bq) . 0 is complex real ext-real Element of REAL
bq |^ (2 * 0) is complex real ext-real Element of REAL
(bq GeoSeq) . (2 * 0) is complex Element of COMPLEX
((- 1) |^ 0) * (bq |^ (2 * 0)) is complex real ext-real Element of REAL
(2 * 0) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * 0) is complex real ext-real Element of REAL
(((- 1) |^ 0) * (bq |^ (2 * 0))) / ((2 * 0) !) is complex real ext-real Element of REAL
((2 * 0) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ 0) * (bq |^ (2 * 0))) * (((2 * 0) !) ") is complex real ext-real set
1 * ((bq GeoSeq) . 0) is complex real ext-real Element of REAL
(1 * ((bq GeoSeq) . 0)) / 1 is complex real ext-real Element of REAL
1 " is non empty complex real ext-real positive non negative set
(1 * ((bq GeoSeq) . 0)) * (1 ") is complex real ext-real set
(Im ((bq * <i>))) . 0 is complex real ext-real Element of REAL
((bq * <i>)) . 0 is complex Element of COMPLEX
Im (((bq * <i>)) . 0) is complex real ext-real Element of REAL
(Im ((bq * <i>))) . 1 is complex real ext-real Element of REAL
((bq * <i>)) . 1 is complex Element of COMPLEX
Im (((bq * <i>)) . 1) is complex real ext-real Element of REAL
0 + (bq * <i>) is complex set
(Partial_Sums (Im ((bq * <i>)))) . ((2 * 0) + 1) is complex real ext-real Element of REAL
(Partial_Sums (Im ((bq * <i>)))) . 0 is complex real ext-real Element of REAL
((Partial_Sums (Im ((bq * <i>)))) . 0) + ((Im ((bq * <i>))) . 1) is complex real ext-real Element of REAL
((Im ((bq * <i>))) . 0) + ((Im ((bq * <i>))) . 1) is complex real ext-real Element of REAL
((bq * <i>)) . (0 + 1) is complex Element of COMPLEX
Im (((bq * <i>)) . (0 + 1)) is complex real ext-real Element of REAL
0 + (Im (((bq * <i>)) . (0 + 1))) is complex real ext-real Element of REAL
(((bq * <i>)) . 0) * (bq * <i>) is complex set
(0 + 1) + (0 * <i>) is complex set
((((bq * <i>)) . 0) * (bq * <i>)) / ((0 + 1) + (0 * <i>)) is complex Element of COMPLEX
((0 + 1) + (0 * <i>)) " is complex set
((((bq * <i>)) . 0) * (bq * <i>)) * (((0 + 1) + (0 * <i>)) ") is complex set
Im (((((bq * <i>)) . 0) * (bq * <i>)) / ((0 + 1) + (0 * <i>))) is complex real ext-real Element of REAL
0 + (Im (((((bq * <i>)) . 0) * (bq * <i>)) / ((0 + 1) + (0 * <i>)))) is complex real ext-real Element of REAL
1 * (bq * <i>) is complex set
(1 * (bq * <i>)) / 1 is complex Element of COMPLEX
(1 * (bq * <i>)) * (1 ") is complex set
Im ((1 * (bq * <i>)) / 1) is complex real ext-real Element of REAL
(Partial_Sums (Re ((bq * <i>)))) . (2 * 0) is complex real ext-real Element of REAL
(Re ((bq * <i>))) . 0 is complex real ext-real Element of REAL
Re (((bq * <i>)) . 0) is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . a is complex real ext-real Element of REAL
2 * a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * a) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (Im ((bq * <i>)))) . ((2 * a) + 1) is complex real ext-real Element of REAL
(Partial_Sums (bq)) . a is complex real ext-real Element of REAL
(Partial_Sums (Re ((bq * <i>)))) . (2 * a) is complex real ext-real Element of REAL
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . (a + 1) is complex real ext-real Element of REAL
2 * (a + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * (a + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (Im ((bq * <i>)))) . ((2 * (a + 1)) + 1) is complex real ext-real Element of REAL
(Partial_Sums (bq)) . (a + 1) is complex real ext-real Element of REAL
(Partial_Sums (Re ((bq * <i>)))) . (2 * (a + 1)) is complex real ext-real Element of REAL
((2 * a) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (Im ((bq * <i>)))) . (((2 * a) + 1) + 1) is complex real ext-real Element of REAL
(Im ((bq * <i>))) . ((2 * (a + 1)) + 1) is complex real ext-real Element of REAL
((Partial_Sums (Im ((bq * <i>)))) . (((2 * a) + 1) + 1)) + ((Im ((bq * <i>))) . ((2 * (a + 1)) + 1)) is complex real ext-real Element of REAL
(Im ((bq * <i>))) . (2 * (a + 1)) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + ((Im ((bq * <i>))) . (2 * (a + 1))) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + ((Im ((bq * <i>))) . (2 * (a + 1)))) + ((Im ((bq * <i>))) . ((2 * (a + 1)) + 1)) is complex real ext-real Element of REAL
((bq * <i>)) . (2 * (a + 1)) is complex Element of COMPLEX
Im (((bq * <i>)) . (2 * (a + 1))) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + (Im (((bq * <i>)) . (2 * (a + 1)))) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Im (((bq * <i>)) . (2 * (a + 1))))) + ((Im ((bq * <i>))) . ((2 * (a + 1)) + 1)) is complex real ext-real Element of REAL
((bq * <i>)) . ((2 * (a + 1)) + 1) is complex Element of COMPLEX
Im (((bq * <i>)) . ((2 * (a + 1)) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Im (((bq * <i>)) . (2 * (a + 1))))) + (Im (((bq * <i>)) . ((2 * (a + 1)) + 1))) is complex real ext-real Element of REAL
(bq * <i>) |^ (2 * (a + 1)) is complex set
(bq * <i>) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((bq * <i>) GeoSeq) . (2 * (a + 1)) is complex Element of COMPLEX
(2 * (a + 1)) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * (a + 1)) is complex real ext-real Element of REAL
((bq * <i>) |^ (2 * (a + 1))) / ((2 * (a + 1)) !) is complex Element of COMPLEX
((2 * (a + 1)) !) " is non empty complex real ext-real positive non negative set
((bq * <i>) |^ (2 * (a + 1))) * (((2 * (a + 1)) !) ") is complex set
Im (((bq * <i>) |^ (2 * (a + 1))) / ((2 * (a + 1)) !)) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + (Im (((bq * <i>) |^ (2 * (a + 1))) / ((2 * (a + 1)) !))) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Im (((bq * <i>) |^ (2 * (a + 1))) / ((2 * (a + 1)) !)))) + (Im (((bq * <i>)) . ((2 * (a + 1)) + 1))) is complex real ext-real Element of REAL
(bq * <i>) |^ ((2 * (a + 1)) + 1) is complex set
((bq * <i>) GeoSeq) . ((2 * (a + 1)) + 1) is complex Element of COMPLEX
((2 * (a + 1)) + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((2 * (a + 1)) + 1) is complex real ext-real Element of REAL
((bq * <i>) |^ ((2 * (a + 1)) + 1)) / (((2 * (a + 1)) + 1) !) is complex Element of COMPLEX
(((2 * (a + 1)) + 1) !) " is non empty complex real ext-real positive non negative set
((bq * <i>) |^ ((2 * (a + 1)) + 1)) * ((((2 * (a + 1)) + 1) !) ") is complex set
Im (((bq * <i>) |^ ((2 * (a + 1)) + 1)) / (((2 * (a + 1)) + 1) !)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Im (((bq * <i>) |^ (2 * (a + 1))) / ((2 * (a + 1)) !)))) + (Im (((bq * <i>) |^ ((2 * (a + 1)) + 1)) / (((2 * (a + 1)) + 1) !))) is complex real ext-real Element of REAL
(- 1) |^ (a + 1) is complex real ext-real Element of REAL
((- 1) GeoSeq) . (a + 1) is complex Element of COMPLEX
bq |^ (2 * (a + 1)) is complex real ext-real Element of REAL
(bq GeoSeq) . (2 * (a + 1)) is complex Element of COMPLEX
((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1))) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) * (((2 * (a + 1)) !) ") is complex real ext-real set
Im ((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + (Im ((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !))) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Im ((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)))) + (Im (((bq * <i>) |^ ((2 * (a + 1)) + 1)) / (((2 * (a + 1)) + 1) !))) is complex real ext-real Element of REAL
bq |^ ((2 * (a + 1)) + 1) is complex real ext-real Element of REAL
(bq GeoSeq) . ((2 * (a + 1)) + 1) is complex Element of COMPLEX
((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) * <i> is complex set
((((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) * <i>) / (((2 * (a + 1)) + 1) !) is complex Element of COMPLEX
((((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) * <i>) * ((((2 * (a + 1)) + 1) !) ") is complex set
Im (((((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) * <i>) / (((2 * (a + 1)) + 1) !)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Im ((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)))) + (Im (((((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) * <i>) / (((2 * (a + 1)) + 1) !))) is complex real ext-real Element of REAL
0 / ((2 * (a + 1)) !) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of REAL
0 * (((2 * (a + 1)) !) ") is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
(0 / ((2 * (a + 1)) !)) * <i> is complex set
((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)) + ((0 / ((2 * (a + 1)) !)) * <i>) is complex set
Im (((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)) + ((0 / ((2 * (a + 1)) !)) * <i>)) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + (Im (((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)) + ((0 / ((2 * (a + 1)) !)) * <i>))) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Im (((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)) + ((0 / ((2 * (a + 1)) !)) * <i>)))) + (Im (((((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) * <i>) / (((2 * (a + 1)) + 1) !))) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + 0 is complex real ext-real Element of REAL
0 / (((2 * (a + 1)) + 1) !) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of REAL
0 * ((((2 * (a + 1)) + 1) !) ") is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
(((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) / (((2 * (a + 1)) + 1) !) is complex real ext-real Element of REAL
(((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) * ((((2 * (a + 1)) + 1) !) ") is complex real ext-real set
((((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) / (((2 * (a + 1)) + 1) !)) * <i> is complex set
(0 / (((2 * (a + 1)) + 1) !)) + (((((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) / (((2 * (a + 1)) + 1) !)) * <i>) is complex set
Im ((0 / (((2 * (a + 1)) + 1) !)) + (((((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) / (((2 * (a + 1)) + 1) !)) * <i>)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + 0) + (Im ((0 / (((2 * (a + 1)) + 1) !)) + (((((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) / (((2 * (a + 1)) + 1) !)) * <i>))) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + (0 / ((2 * (a + 1)) !)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (0 / ((2 * (a + 1)) !))) + ((((- 1) |^ (a + 1)) * (bq |^ ((2 * (a + 1)) + 1))) / (((2 * (a + 1)) + 1) !)) is complex real ext-real Element of REAL
(bq) . (a + 1) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + ((bq) . (a + 1)) is complex real ext-real Element of REAL
(Partial_Sums (Re ((bq * <i>)))) . ((2 * a) + 1) is complex real ext-real Element of REAL
(Re ((bq * <i>))) . (((2 * a) + 1) + 1) is complex real ext-real Element of REAL
((Partial_Sums (Re ((bq * <i>)))) . ((2 * a) + 1)) + ((Re ((bq * <i>))) . (((2 * a) + 1) + 1)) is complex real ext-real Element of REAL
(Re ((bq * <i>))) . ((2 * a) + 1) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + ((Re ((bq * <i>))) . ((2 * a) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + ((Re ((bq * <i>))) . ((2 * a) + 1))) + ((Re ((bq * <i>))) . (((2 * a) + 1) + 1)) is complex real ext-real Element of REAL
((bq * <i>)) . ((2 * a) + 1) is complex Element of COMPLEX
Re (((bq * <i>)) . ((2 * a) + 1)) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + (Re (((bq * <i>)) . ((2 * a) + 1))) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Re (((bq * <i>)) . ((2 * a) + 1)))) + ((Re ((bq * <i>))) . (((2 * a) + 1) + 1)) is complex real ext-real Element of REAL
((bq * <i>)) . (((2 * a) + 1) + 1) is complex Element of COMPLEX
Re (((bq * <i>)) . (((2 * a) + 1) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Re (((bq * <i>)) . ((2 * a) + 1)))) + (Re (((bq * <i>)) . (((2 * a) + 1) + 1))) is complex real ext-real Element of REAL
(bq * <i>) |^ ((2 * a) + 1) is complex set
((bq * <i>) GeoSeq) . ((2 * a) + 1) is complex Element of COMPLEX
((2 * a) + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((2 * a) + 1) is complex real ext-real Element of REAL
((bq * <i>) |^ ((2 * a) + 1)) / (((2 * a) + 1) !) is complex Element of COMPLEX
(((2 * a) + 1) !) " is non empty complex real ext-real positive non negative set
((bq * <i>) |^ ((2 * a) + 1)) * ((((2 * a) + 1) !) ") is complex set
Re (((bq * <i>) |^ ((2 * a) + 1)) / (((2 * a) + 1) !)) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + (Re (((bq * <i>) |^ ((2 * a) + 1)) / (((2 * a) + 1) !))) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Re (((bq * <i>) |^ ((2 * a) + 1)) / (((2 * a) + 1) !)))) + (Re (((bq * <i>)) . (((2 * a) + 1) + 1))) is complex real ext-real Element of REAL
(((2 * a) + 1) + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (((2 * a) + 1) + 1) is complex real ext-real Element of REAL
((bq * <i>) |^ (2 * (a + 1))) / ((((2 * a) + 1) + 1) !) is complex Element of COMPLEX
((((2 * a) + 1) + 1) !) " is non empty complex real ext-real positive non negative set
((bq * <i>) |^ (2 * (a + 1))) * (((((2 * a) + 1) + 1) !) ") is complex set
Re (((bq * <i>) |^ (2 * (a + 1))) / ((((2 * a) + 1) + 1) !)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Re (((bq * <i>) |^ ((2 * a) + 1)) / (((2 * a) + 1) !)))) + (Re (((bq * <i>) |^ (2 * (a + 1))) / ((((2 * a) + 1) + 1) !))) is complex real ext-real Element of REAL
(- 1) |^ a is complex real ext-real Element of REAL
((- 1) GeoSeq) . a is complex Element of COMPLEX
bq |^ ((2 * a) + 1) is complex real ext-real Element of REAL
(bq GeoSeq) . ((2 * a) + 1) is complex Element of COMPLEX
((- 1) |^ a) * (bq |^ ((2 * a) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ a) * (bq |^ ((2 * a) + 1))) * <i> is complex set
((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) * <i>) / (((2 * a) + 1) !) is complex Element of COMPLEX
((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) * <i>) * ((((2 * a) + 1) !) ") is complex set
Re (((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) * <i>) / (((2 * a) + 1) !)) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + (Re (((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) * <i>) / (((2 * a) + 1) !))) is complex real ext-real Element of REAL
Re (((bq * <i>) |^ (2 * (a + 1))) / ((2 * (a + 1)) !)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Re (((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) * <i>) / (((2 * a) + 1) !)))) + (Re (((bq * <i>) |^ (2 * (a + 1))) / ((2 * (a + 1)) !))) is complex real ext-real Element of REAL
Re ((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Re (((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) * <i>) / (((2 * a) + 1) !)))) + (Re ((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !))) is complex real ext-real Element of REAL
0 / (((2 * a) + 1) !) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of REAL
0 * ((((2 * a) + 1) !) ") is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
(((- 1) |^ a) * (bq |^ ((2 * a) + 1))) / (((2 * a) + 1) !) is complex real ext-real Element of REAL
(((- 1) |^ a) * (bq |^ ((2 * a) + 1))) * ((((2 * a) + 1) !) ") is complex real ext-real set
((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) / (((2 * a) + 1) !)) * <i> is complex set
(0 / (((2 * a) + 1) !)) + (((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) / (((2 * a) + 1) !)) * <i>) is complex set
Re ((0 / (((2 * a) + 1) !)) + (((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) / (((2 * a) + 1) !)) * <i>)) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + (Re ((0 / (((2 * a) + 1) !)) + (((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) / (((2 * a) + 1) !)) * <i>))) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (Re ((0 / (((2 * a) + 1) !)) + (((((- 1) |^ a) * (bq |^ ((2 * a) + 1))) / (((2 * a) + 1) !)) * <i>)))) + (Re ((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !))) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + (0 / (((2 * a) + 1) !)) is complex real ext-real Element of REAL
Re (((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)) + ((0 / ((2 * (a + 1)) !)) * <i>)) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (0 / (((2 * a) + 1) !))) + (Re (((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)) + ((0 / ((2 * (a + 1)) !)) * <i>))) is complex real ext-real Element of REAL
(((Partial_Sums (bq)) . a) + (0 / (((2 * a) + 1) !))) + ((((- 1) |^ (a + 1)) * (bq |^ (2 * (a + 1)))) / ((2 * (a + 1)) !)) is complex real ext-real Element of REAL
(bq) . (a + 1) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . a) + ((bq) . (a + 1)) is complex real ext-real Element of REAL
bq is complex real ext-real Element of REAL
(bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (bq) is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
Im (Sum ((bq * <i>))) is complex real ext-real Element of REAL
(bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (bq) is complex real ext-real Element of REAL
Re (Sum ((bq * <i>))) is complex real ext-real Element of REAL
Re ((bq * <i>)) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
Sum (Re ((bq * <i>))) is complex real ext-real Element of REAL
Im ((bq * <i>)) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
Sum (Im ((bq * <i>))) is complex real ext-real Element of REAL
(Sum (Im ((bq * <i>)))) * <i> is complex set
(Sum (Re ((bq * <i>)))) + ((Sum (Im ((bq * <i>)))) * <i>) is complex set
Partial_Sums (Re ((bq * <i>))) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim (Partial_Sums (Re ((bq * <i>)))) is complex real ext-real Element of REAL
Partial_Sums (Im ((bq * <i>))) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim (Partial_Sums (Im ((bq * <i>)))) is complex real ext-real Element of REAL
bq is complex real ext-real set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . th1 is complex real ext-real Element of REAL
((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
- (Re (Sum ((bq * <i>)))) is complex real ext-real set
((Partial_Sums (bq)) . th1) + (- (Re (Sum ((bq * <i>))))) is complex real ext-real set
abs (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) * (Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) is complex real ext-real set
Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) * (Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) is complex real ext-real set
((Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2)) is complex real ext-real Element of REAL
2 * th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (Re ((bq * <i>)))) . (2 * th1) is complex real ext-real Element of REAL
((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) + (- (Re (Sum ((bq * <i>))))) is complex real ext-real set
abs (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
Re (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))))) * (Re (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))))) is complex real ext-real set
Im (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))))) * (Im (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))))) is complex real ext-real set
((Re (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (Re ((bq * <i>)))) . (2 * th1)) - (Re (Sum ((bq * <i>)))))) ^2)) is complex real ext-real Element of REAL
th1 + th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . th1 is complex real ext-real Element of REAL
((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . th1) + (- (Re (Sum ((bq * <i>))))) is complex real ext-real set
abs (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) * (Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) is complex real ext-real set
Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) * (Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) is complex real ext-real set
((Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (bq)) . th1) - (Re (Sum ((bq * <i>)))))) ^2)) is complex real ext-real Element of REAL
lim (Partial_Sums (bq)) is complex real ext-real Element of REAL
bq is complex real ext-real set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . th1 is complex real ext-real Element of REAL
((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
- (Im (Sum ((bq * <i>)))) is complex real ext-real set
((Partial_Sums (bq)) . th1) + (- (Im (Sum ((bq * <i>))))) is complex real ext-real set
abs (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) * (Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) is complex real ext-real set
Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) * (Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) is complex real ext-real set
((Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2)) is complex real ext-real Element of REAL
2 * th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * th1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1) is complex real ext-real Element of REAL
((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) + (- (Im (Sum ((bq * <i>))))) is complex real ext-real set
abs (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
Re (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))))) * (Re (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))))) is complex real ext-real set
Im (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))))) * (Im (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))))) is complex real ext-real set
((Re (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (Im ((bq * <i>)))) . ((2 * th1) + 1)) - (Im (Sum ((bq * <i>)))))) ^2)) is complex real ext-real Element of REAL
th1 + th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (bq)) . th1 is complex real ext-real Element of REAL
((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))) is complex real ext-real Element of REAL
((Partial_Sums (bq)) . th1) + (- (Im (Sum ((bq * <i>))))) is complex real ext-real set
abs (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) * (Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) is complex real ext-real set
Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>))))) is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2 is complex real ext-real Element of REAL
(Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) * (Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) is complex real ext-real set
((Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2) + ((Im (((Partial_Sums (bq)) . th1) - (Im (Sum ((bq * <i>)))))) ^2)) is complex real ext-real Element of REAL
lim (Partial_Sums (bq)) is complex real ext-real Element of REAL
bq is complex real ext-real set
() . bq is complex real ext-real Element of REAL
(bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (bq) is complex real ext-real Element of REAL
() . bq is complex real ext-real Element of REAL
(bq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (bq) is complex real ext-real Element of REAL
bq is complex real ext-real Element of REAL
() . bq is complex real ext-real Element of REAL
bq * <i> is complex set
((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((bq * <i>)) is complex Element of COMPLEX
Partial_Sums ((bq * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((bq * <i>))) is complex Element of COMPLEX
Im (Sum ((bq * <i>))) is complex real ext-real Element of REAL
() . bq is complex real ext-real Element of REAL
Re (Sum ((bq * <i>))) is complex real ext-real Element of REAL
a is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim a is complex real ext-real Element of REAL
bq is complex real ext-real set
bq is complex real ext-real set
bq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . bq is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . a is complex real ext-real Element of REAL
2 * a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
2 * bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * a) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * bq) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
bq . bq is complex real ext-real Element of REAL
2 * bq is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * bq) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
2 * a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(- 1) |^ (2 * a) is complex real ext-real Element of REAL
(- 1) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((- 1) GeoSeq) . (2 * a) is complex Element of COMPLEX
(2 * a) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(- 1) |^ ((2 * a) + 1) is complex real ext-real Element of REAL
((- 1) GeoSeq) . ((2 * a) + 1) is complex Element of COMPLEX
b is complex real ext-real set
b |^ (2 * a) is complex real ext-real set
b GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(b GeoSeq) . (2 * a) is complex Element of COMPLEX
b |^ a is complex real ext-real set
(b GeoSeq) . a is complex Element of COMPLEX
(b |^ a) |^ 2 is complex real ext-real set
(b |^ a) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((b |^ a) GeoSeq) . 2 is complex Element of COMPLEX
1 |^ (2 * a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
1 GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(1 GeoSeq) . (2 * a) is complex Element of COMPLEX
((- 1) |^ (2 * a)) * (- 1) is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . a is complex real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . b is complex real ext-real Element of REAL
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 . th2 is complex real ext-real Element of REAL
th2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 . (th2 + 1) is complex real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 . (0 + 1) is complex real ext-real Element of REAL
(0 + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (0 + 1) is complex real ext-real Element of REAL
(0 !) * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 . 0 is complex real ext-real Element of REAL
PL is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 . PL is complex real ext-real Element of REAL
PL + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 . (PL + 1) is complex real ext-real Element of REAL
(PL + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 . ((PL + 1) + 1) is complex real ext-real Element of REAL
((PL + 1) + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((PL + 1) + 1) is complex real ext-real Element of REAL
(PL + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (PL + 1) is complex real ext-real Element of REAL
((PL + 1) !) * ((PL + 1) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(th1 . (PL + 1)) * ((PL + 1) + 1) is complex real ext-real Element of REAL
th1 . a is complex real ext-real Element of REAL
th1 . b is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 is complex real ext-real set
th1 |^ b is complex real ext-real set
th1 GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(th1 GeoSeq) . b is complex Element of COMPLEX
th1 |^ a is complex real ext-real set
(th1 GeoSeq) . a is complex Element of COMPLEX
th1 GeoSeq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(th1 GeoSeq) . (th2 + 1) is complex real ext-real Element of REAL
(th1 GeoSeq) . th2 is complex real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(th1 GeoSeq) . (0 + 1) is complex real ext-real Element of REAL
(th1 GeoSeq) . 0 is complex real ext-real Element of REAL
((th1 GeoSeq) . 0) * th1 is complex real ext-real Element of REAL
1 * th1 is complex real ext-real Element of REAL
PL is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
PL + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(th1 GeoSeq) . (PL + 1) is complex real ext-real Element of REAL
(th1 GeoSeq) . PL is complex real ext-real Element of REAL
(PL + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(th1 GeoSeq) . ((PL + 1) + 1) is complex real ext-real Element of REAL
((th1 GeoSeq) . (PL + 1)) * th1 is complex real ext-real Element of REAL
((th1 GeoSeq) . PL) * th1 is complex real ext-real Element of REAL
(th1 GeoSeq) . b is complex real ext-real Element of REAL
(th1 GeoSeq) . a is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b |^ a is complex real ext-real Element of REAL
b GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(b GeoSeq) . a is complex Element of COMPLEX
a ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . a is complex real ext-real Element of REAL
(b |^ a) / (a !) is complex real ext-real Element of REAL
(a !) " is non empty complex real ext-real positive non negative set
(b |^ a) * ((a !) ") is complex real ext-real set
a is complex real ext-real set
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
Im (Sum (a)) is complex real ext-real Element of REAL
Im (a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
Partial_Sums (Im (a)) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (Im (a))) . b is complex real ext-real Element of REAL
(Partial_Sums (Im (a))) . 0 is complex real ext-real Element of REAL
(Im (a)) . 0 is complex real ext-real Element of REAL
(a) . 0 is complex Element of COMPLEX
Im ((a) . 0) is complex real ext-real Element of REAL
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (Im (a))) . th1 is complex real ext-real Element of REAL
th1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (Im (a))) . (th1 + 1) is complex real ext-real Element of REAL
(Im (a)) . (th1 + 1) is complex real ext-real Element of REAL
0 + ((Im (a)) . (th1 + 1)) is complex real ext-real Element of REAL
(a) . (th1 + 1) is complex Element of COMPLEX
Im ((a) . (th1 + 1)) is complex real ext-real Element of REAL
a |^ (th1 + 1) is complex real ext-real set
a GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a GeoSeq) . (th1 + 1) is complex Element of COMPLEX
(th1 + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (th1 + 1) is complex real ext-real Element of REAL
(a |^ (th1 + 1)) / ((th1 + 1) !) is complex real ext-real Element of REAL
((th1 + 1) !) " is non empty complex real ext-real positive non negative set
(a |^ (th1 + 1)) * (((th1 + 1) !) ") is complex real ext-real set
Im ((a |^ (th1 + 1)) / ((th1 + 1) !)) is complex real ext-real Element of REAL
((a |^ (th1 + 1)) / ((th1 + 1) !)) + (0 * <i>) is complex set
Im (((a |^ (th1 + 1)) / ((th1 + 1) !)) + (0 * <i>)) is complex real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative set
(Partial_Sums (Im (a))) . b is complex real ext-real Element of REAL
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative set
(Partial_Sums (Im (a))) . th1 is complex real ext-real Element of REAL
lim (Partial_Sums (Im (a))) is complex real ext-real Element of REAL
(Partial_Sums (Im (a))) . 0 is complex real ext-real Element of REAL
Re (a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
Sum (Re (a)) is complex real ext-real Element of REAL
Sum (Im (a)) is complex real ext-real Element of REAL
(Sum (Im (a))) * <i> is complex set
(Sum (Re (a))) + ((Sum (Im (a))) * <i>) is complex set
Im ((Sum (Re (a))) + ((Sum (Im (a))) * <i>)) is complex real ext-real Element of REAL
() . 1 is complex real ext-real Element of REAL
() . 1 is complex real ext-real Element of REAL
(1) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (1) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (1) is complex real ext-real Element of REAL
bq is Relation-like NAT -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K19(K20(NAT,NAT))
(Partial_Sums (1)) * bq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued subsequence of Partial_Sums (1)
lim ((Partial_Sums (1)) * bq) is complex real ext-real Element of REAL
lim (Partial_Sums (1)) is complex real ext-real Element of REAL
1 / 2 is non empty complex real ext-real positive non negative Element of REAL
2 " is non empty complex real ext-real positive non negative set
1 * (2 ") is non empty complex real ext-real positive non negative set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums (1)) * bq) . a is complex real ext-real Element of REAL
((Partial_Sums (1)) * bq) . 0 is complex real ext-real Element of REAL
bq . 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (1)) . (bq . 0) is complex real ext-real Element of REAL
2 * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() V62() V63() V98() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (1)) . ((2 * 0) + 1) is complex real ext-real Element of REAL
(Partial_Sums (1)) . 0 is complex real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1) . (0 + 1) is complex real ext-real Element of REAL
((Partial_Sums (1)) . 0) + ((1) . (0 + 1)) is complex real ext-real Element of REAL
(1) . 0 is complex real ext-real Element of REAL
((1) . 0) + ((1) . (0 + 1)) is complex real ext-real Element of REAL
(- 1) |^ 0 is complex real ext-real Element of REAL
(- 1) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((- 1) GeoSeq) . 0 is complex Element of COMPLEX
1 |^ (2 * 0) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
1 GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(1 GeoSeq) . (2 * 0) is complex Element of COMPLEX
((- 1) |^ 0) * (1 |^ (2 * 0)) is complex real ext-real Element of REAL
(2 * 0) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * 0) is complex real ext-real Element of REAL
(((- 1) |^ 0) * (1 |^ (2 * 0))) / ((2 * 0) !) is complex real ext-real Element of REAL
((2 * 0) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ 0) * (1 |^ (2 * 0))) * (((2 * 0) !) ") is complex real ext-real set
(1) . 1 is complex real ext-real Element of REAL
((((- 1) |^ 0) * (1 |^ (2 * 0))) / ((2 * 0) !)) + ((1) . 1) is complex real ext-real Element of REAL
(- 1) |^ 1 is complex real ext-real Element of REAL
((- 1) GeoSeq) . 1 is complex Element of COMPLEX
2 * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
1 |^ (2 * 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 GeoSeq) . (2 * 1) is complex Element of COMPLEX
((- 1) |^ 1) * (1 |^ (2 * 1)) is complex real ext-real Element of REAL
(2 * 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * 1) is complex real ext-real Element of REAL
(((- 1) |^ 1) * (1 |^ (2 * 1))) / ((2 * 1) !) is complex real ext-real Element of REAL
((2 * 1) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ 1) * (1 |^ (2 * 1))) * (((2 * 1) !) ") is complex real ext-real set
((((- 1) |^ 0) * (1 |^ (2 * 0))) / ((2 * 0) !)) + ((((- 1) |^ 1) * (1 |^ (2 * 1))) / ((2 * 1) !)) is complex real ext-real Element of REAL
1 * (1 |^ (2 * 0)) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 * (1 |^ (2 * 0))) / ((2 * 0) !) is complex real ext-real non negative Element of REAL
(1 * (1 |^ (2 * 0))) * (((2 * 0) !) ") is complex real ext-real non negative set
((1 * (1 |^ (2 * 0))) / ((2 * 0) !)) + ((((- 1) |^ 1) * (1 |^ (2 * 1))) / ((2 * 1) !)) is complex real ext-real Element of REAL
1 / 1 is non empty complex real ext-real positive non negative Element of REAL
1 " is non empty complex real ext-real positive non negative set
1 * (1 ") is non empty complex real ext-real positive non negative set
(1 / 1) + ((((- 1) |^ 1) * (1 |^ (2 * 1))) / ((2 * 1) !)) is complex real ext-real Element of REAL
(- 1) * (1 |^ (2 * 1)) is complex real ext-real non positive Element of REAL
((- 1) * (1 |^ (2 * 1))) / ((2 * 1) !) is complex real ext-real non positive Element of REAL
((- 1) * (1 |^ (2 * 1))) * (((2 * 1) !) ") is complex real ext-real non positive set
1 + (((- 1) * (1 |^ (2 * 1))) / ((2 * 1) !)) is complex real ext-real Element of REAL
(- 1) * 1 is non empty complex real ext-real non positive negative Element of REAL
((- 1) * 1) / ((2 * 1) !) is non empty complex real ext-real non positive negative Element of REAL
((- 1) * 1) * (((2 * 1) !) ") is non empty complex real ext-real non positive negative set
1 + (((- 1) * 1) / ((2 * 1) !)) is complex real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 !) * (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(- 1) / ((1 !) * (1 + 1)) is non empty complex real ext-real non positive negative Element of REAL
((1 !) * (1 + 1)) " is non empty complex real ext-real positive non negative set
(- 1) * (((1 !) * (1 + 1)) ") is non empty complex real ext-real non positive negative set
1 + ((- 1) / ((1 !) * (1 + 1))) is complex real ext-real Element of REAL
(0 !) * (0 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((0 !) * (0 + 1)) * 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(- 1) / (((0 !) * (0 + 1)) * 2) is non empty complex real ext-real non positive negative Element of REAL
(((0 !) * (0 + 1)) * 2) " is non empty complex real ext-real positive non negative set
(- 1) * ((((0 !) * (0 + 1)) * 2) ") is non empty complex real ext-real non positive negative set
1 + ((- 1) / (((0 !) * (0 + 1)) * 2)) is complex real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums (1)) * bq) . b is complex real ext-real Element of REAL
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums (1)) * bq) . (b + 1) is complex real ext-real Element of REAL
bq . (b + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (1)) . (bq . (b + 1)) is complex real ext-real Element of REAL
2 * (b + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * (b + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (1)) . ((2 * (b + 1)) + 1) is complex real ext-real Element of REAL
2 * b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * b) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
((2 * b) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (1)) . (((2 * b) + 1) + 1) is complex real ext-real Element of REAL
(1) . ((2 * (b + 1)) + 1) is complex real ext-real Element of REAL
((Partial_Sums (1)) . (((2 * b) + 1) + 1)) + ((1) . ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
(Partial_Sums (1)) . ((2 * b) + 1) is complex real ext-real Element of REAL
(1) . (((2 * b) + 1) + 1) is complex real ext-real Element of REAL
((Partial_Sums (1)) . ((2 * b) + 1)) + ((1) . (((2 * b) + 1) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (1)) . ((2 * b) + 1)) + ((1) . (((2 * b) + 1) + 1))) + ((1) . ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
bq . b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (1)) . (bq . b) is complex real ext-real Element of REAL
((Partial_Sums (1)) . (bq . b)) + ((1) . (((2 * b) + 1) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (1)) . (bq . b)) + ((1) . (((2 * b) + 1) + 1))) + ((1) . ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (1)) * bq) . b) + ((1) . (((2 * b) + 1) + 1)) is complex real ext-real Element of REAL
((((Partial_Sums (1)) * bq) . b) + ((1) . (((2 * b) + 1) + 1))) + ((1) . ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (1)) * bq) . (b + 1)) - (((Partial_Sums (1)) * bq) . b) is complex real ext-real Element of REAL
- (((Partial_Sums (1)) * bq) . b) is complex real ext-real set
(((Partial_Sums (1)) * bq) . (b + 1)) + (- (((Partial_Sums (1)) * bq) . b)) is complex real ext-real set
((1) . (((2 * b) + 1) + 1)) + ((1) . ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
(- 1) |^ (2 * (b + 1)) is complex real ext-real Element of REAL
((- 1) GeoSeq) . (2 * (b + 1)) is complex Element of COMPLEX
2 * (2 * (b + 1)) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
1 |^ (2 * (2 * (b + 1))) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 GeoSeq) . (2 * (2 * (b + 1))) is complex Element of COMPLEX
((- 1) |^ (2 * (b + 1))) * (1 |^ (2 * (2 * (b + 1)))) is complex real ext-real Element of REAL
(2 * (2 * (b + 1))) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * (2 * (b + 1))) is complex real ext-real Element of REAL
(((- 1) |^ (2 * (b + 1))) * (1 |^ (2 * (2 * (b + 1))))) / ((2 * (2 * (b + 1))) !) is complex real ext-real Element of REAL
((2 * (2 * (b + 1))) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ (2 * (b + 1))) * (1 |^ (2 * (2 * (b + 1))))) * (((2 * (2 * (b + 1))) !) ") is complex real ext-real set
1 * (1 |^ (2 * (2 * (b + 1)))) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 * (1 |^ (2 * (2 * (b + 1))))) / ((2 * (2 * (b + 1))) !) is complex real ext-real non negative Element of REAL
(1 * (1 |^ (2 * (2 * (b + 1))))) * (((2 * (2 * (b + 1))) !) ") is complex real ext-real non negative set
1 / ((2 * (2 * (b + 1))) !) is non empty complex real ext-real positive non negative Element of REAL
1 * (((2 * (2 * (b + 1))) !) ") is non empty complex real ext-real positive non negative set
(- 1) |^ ((2 * (b + 1)) + 1) is complex real ext-real Element of REAL
((- 1) GeoSeq) . ((2 * (b + 1)) + 1) is complex Element of COMPLEX
2 * ((2 * (b + 1)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
1 |^ (2 * ((2 * (b + 1)) + 1)) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 GeoSeq) . (2 * ((2 * (b + 1)) + 1)) is complex Element of COMPLEX
((- 1) |^ ((2 * (b + 1)) + 1)) * (1 |^ (2 * ((2 * (b + 1)) + 1))) is complex real ext-real Element of REAL
(2 * ((2 * (b + 1)) + 1)) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ ((2 * (b + 1)) + 1)) * (1 |^ (2 * ((2 * (b + 1)) + 1)))) / ((2 * ((2 * (b + 1)) + 1)) !) is complex real ext-real Element of REAL
((2 * ((2 * (b + 1)) + 1)) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ ((2 * (b + 1)) + 1)) * (1 |^ (2 * ((2 * (b + 1)) + 1)))) * (((2 * ((2 * (b + 1)) + 1)) !) ") is complex real ext-real set
(- 1) * (1 |^ (2 * ((2 * (b + 1)) + 1))) is complex real ext-real non positive Element of REAL
((- 1) * (1 |^ (2 * ((2 * (b + 1)) + 1)))) / ((2 * ((2 * (b + 1)) + 1)) !) is complex real ext-real non positive Element of REAL
((- 1) * (1 |^ (2 * ((2 * (b + 1)) + 1)))) * (((2 * ((2 * (b + 1)) + 1)) !) ") is complex real ext-real non positive set
((- 1) * 1) / ((2 * ((2 * (b + 1)) + 1)) !) is non empty complex real ext-real non positive negative Element of REAL
((- 1) * 1) * (((2 * ((2 * (b + 1)) + 1)) !) ") is non empty complex real ext-real non positive negative set
(- 1) / ((2 * ((2 * (b + 1)) + 1)) !) is non empty complex real ext-real non positive negative Element of REAL
(- 1) * (((2 * ((2 * (b + 1)) + 1)) !) ") is non empty complex real ext-real non positive negative set
1 / ((2 * ((2 * (b + 1)) + 1)) !) is non empty complex real ext-real positive non negative Element of REAL
1 * (((2 * ((2 * (b + 1)) + 1)) !) ") is non empty complex real ext-real positive non negative set
(1 / ((2 * (2 * (b + 1))) !)) - (1 / ((2 * ((2 * (b + 1)) + 1)) !)) is complex real ext-real Element of REAL
- (1 / ((2 * ((2 * (b + 1)) + 1)) !)) is non empty complex real ext-real non positive negative set
(1 / ((2 * (2 * (b + 1))) !)) + (- (1 / ((2 * ((2 * (b + 1)) + 1)) !))) is complex real ext-real set
(1) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (1) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (1) is complex real ext-real Element of REAL
(Partial_Sums (1)) * bq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued subsequence of Partial_Sums (1)
lim ((Partial_Sums (1)) * bq) is complex real ext-real Element of REAL
lim (Partial_Sums (1)) is complex real ext-real Element of REAL
5 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
6 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
5 / 6 is non empty complex real ext-real positive non negative Element of REAL
6 " is non empty complex real ext-real positive non negative set
5 * (6 ") is non empty complex real ext-real positive non negative set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums (1)) * bq) . a is complex real ext-real Element of REAL
((Partial_Sums (1)) * bq) . 0 is complex real ext-real Element of REAL
bq . 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (1)) . (bq . 0) is complex real ext-real Element of REAL
2 * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() V62() V63() V98() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (1)) . ((2 * 0) + 1) is complex real ext-real Element of REAL
(Partial_Sums (1)) . 0 is complex real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1) . (0 + 1) is complex real ext-real Element of REAL
((Partial_Sums (1)) . 0) + ((1) . (0 + 1)) is complex real ext-real Element of REAL
(1) . 0 is complex real ext-real Element of REAL
((1) . 0) + ((1) . (0 + 1)) is complex real ext-real Element of REAL
(- 1) |^ 0 is complex real ext-real Element of REAL
(- 1) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((- 1) GeoSeq) . 0 is complex Element of COMPLEX
1 |^ ((2 * 0) + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
1 GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(1 GeoSeq) . ((2 * 0) + 1) is complex Element of COMPLEX
((- 1) |^ 0) * (1 |^ ((2 * 0) + 1)) is complex real ext-real Element of REAL
((2 * 0) + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((2 * 0) + 1) is complex real ext-real Element of REAL
(((- 1) |^ 0) * (1 |^ ((2 * 0) + 1))) / (((2 * 0) + 1) !) is complex real ext-real Element of REAL
(((2 * 0) + 1) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ 0) * (1 |^ ((2 * 0) + 1))) * ((((2 * 0) + 1) !) ") is complex real ext-real set
(1) . 1 is complex real ext-real Element of REAL
((((- 1) |^ 0) * (1 |^ ((2 * 0) + 1))) / (((2 * 0) + 1) !)) + ((1) . 1) is complex real ext-real Element of REAL
(- 1) |^ 1 is complex real ext-real Element of REAL
((- 1) GeoSeq) . 1 is complex Element of COMPLEX
2 * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
1 |^ ((2 * 1) + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 GeoSeq) . ((2 * 1) + 1) is complex Element of COMPLEX
((- 1) |^ 1) * (1 |^ ((2 * 1) + 1)) is complex real ext-real Element of REAL
((2 * 1) + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((2 * 1) + 1) is complex real ext-real Element of REAL
(((- 1) |^ 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !) is complex real ext-real Element of REAL
(((2 * 1) + 1) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ 1) * (1 |^ ((2 * 1) + 1))) * ((((2 * 1) + 1) !) ") is complex real ext-real set
((((- 1) |^ 0) * (1 |^ ((2 * 0) + 1))) / (((2 * 0) + 1) !)) + ((((- 1) |^ 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !)) is complex real ext-real Element of REAL
1 * (1 |^ ((2 * 0) + 1)) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 * (1 |^ ((2 * 0) + 1))) / (((2 * 0) + 1) !) is complex real ext-real non negative Element of REAL
(1 * (1 |^ ((2 * 0) + 1))) * ((((2 * 0) + 1) !) ") is complex real ext-real non negative set
((1 * (1 |^ ((2 * 0) + 1))) / (((2 * 0) + 1) !)) + ((((- 1) |^ 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !)) is complex real ext-real Element of REAL
(0 + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (0 + 1) is complex real ext-real Element of REAL
1 / ((0 + 1) !) is non empty complex real ext-real positive non negative Element of REAL
((0 + 1) !) " is non empty complex real ext-real positive non negative set
1 * (((0 + 1) !) ") is non empty complex real ext-real positive non negative set
(1 / ((0 + 1) !)) + ((((- 1) |^ 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !)) is complex real ext-real Element of REAL
(0 !) * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
1 / ((0 !) * 1) is non empty complex real ext-real positive non negative Element of REAL
((0 !) * 1) " is non empty complex real ext-real positive non negative set
1 * (((0 !) * 1) ") is non empty complex real ext-real positive non negative set
(1 / ((0 !) * 1)) + ((((- 1) |^ 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !)) is complex real ext-real Element of REAL
(- 1) * (1 |^ ((2 * 1) + 1)) is complex real ext-real non positive Element of REAL
((- 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !) is complex real ext-real non positive Element of REAL
((- 1) * (1 |^ ((2 * 1) + 1))) * ((((2 * 1) + 1) !) ") is complex real ext-real non positive set
1 + (((- 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !)) is complex real ext-real Element of REAL
(- 1) * 1 is non empty complex real ext-real non positive negative Element of REAL
((- 1) * 1) / (((2 * 1) + 1) !) is non empty complex real ext-real non positive negative Element of REAL
((- 1) * 1) * ((((2 * 1) + 1) !) ") is non empty complex real ext-real non positive negative set
1 + (((- 1) * 1) / (((2 * 1) + 1) !)) is complex real ext-real Element of REAL
(2 * 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * 1) is complex real ext-real Element of REAL
((2 * 1) !) * ((2 * 1) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(- 1) / (((2 * 1) !) * ((2 * 1) + 1)) is non empty complex real ext-real non positive negative Element of REAL
(((2 * 1) !) * ((2 * 1) + 1)) " is non empty complex real ext-real positive non negative set
(- 1) * ((((2 * 1) !) * ((2 * 1) + 1)) ") is non empty complex real ext-real non positive negative set
1 + ((- 1) / (((2 * 1) !) * ((2 * 1) + 1))) is complex real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 !) * (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
3 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((1 !) * (1 + 1)) * 3 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(- 1) / (((1 !) * (1 + 1)) * 3) is non empty complex real ext-real non positive negative Element of REAL
(((1 !) * (1 + 1)) * 3) " is non empty complex real ext-real positive non negative set
(- 1) * ((((1 !) * (1 + 1)) * 3) ") is non empty complex real ext-real non positive negative set
1 + ((- 1) / (((1 !) * (1 + 1)) * 3)) is complex real ext-real Element of REAL
2 * 3 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
((0 + 1) !) * (2 * 3) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(- 1) / (((0 + 1) !) * (2 * 3)) is non empty complex real ext-real non positive negative Element of REAL
(((0 + 1) !) * (2 * 3)) " is non empty complex real ext-real positive non negative set
(- 1) * ((((0 + 1) !) * (2 * 3)) ") is non empty complex real ext-real non positive negative set
1 + ((- 1) / (((0 + 1) !) * (2 * 3))) is complex real ext-real Element of REAL
1 * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 * 1) * 6 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(- 1) / ((1 * 1) * 6) is non empty complex real ext-real non positive negative Element of REAL
((1 * 1) * 6) " is non empty complex real ext-real positive non negative set
(- 1) * (((1 * 1) * 6) ") is non empty complex real ext-real non positive negative set
1 + ((- 1) / ((1 * 1) * 6)) is complex real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums (1)) * bq) . b is complex real ext-real Element of REAL
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums (1)) * bq) . (b + 1) is complex real ext-real Element of REAL
bq . (b + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (1)) . (bq . (b + 1)) is complex real ext-real Element of REAL
2 * (b + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * (b + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (1)) . ((2 * (b + 1)) + 1) is complex real ext-real Element of REAL
2 * b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * b) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
((2 * b) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (1)) . (((2 * b) + 1) + 1) is complex real ext-real Element of REAL
(1) . ((2 * (b + 1)) + 1) is complex real ext-real Element of REAL
((Partial_Sums (1)) . (((2 * b) + 1) + 1)) + ((1) . ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
(Partial_Sums (1)) . ((2 * b) + 1) is complex real ext-real Element of REAL
(1) . (((2 * b) + 1) + 1) is complex real ext-real Element of REAL
((Partial_Sums (1)) . ((2 * b) + 1)) + ((1) . (((2 * b) + 1) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (1)) . ((2 * b) + 1)) + ((1) . (((2 * b) + 1) + 1))) + ((1) . ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
bq . b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (1)) . (bq . b) is complex real ext-real Element of REAL
((Partial_Sums (1)) . (bq . b)) + ((1) . (((2 * b) + 1) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (1)) . (bq . b)) + ((1) . (((2 * b) + 1) + 1))) + ((1) . ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (1)) * bq) . b) + ((1) . (((2 * b) + 1) + 1)) is complex real ext-real Element of REAL
((((Partial_Sums (1)) * bq) . b) + ((1) . (((2 * b) + 1) + 1))) + ((1) . ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (1)) * bq) . (b + 1)) - (((Partial_Sums (1)) * bq) . b) is complex real ext-real Element of REAL
- (((Partial_Sums (1)) * bq) . b) is complex real ext-real set
(((Partial_Sums (1)) * bq) . (b + 1)) + (- (((Partial_Sums (1)) * bq) . b)) is complex real ext-real set
((1) . (((2 * b) + 1) + 1)) + ((1) . ((2 * (b + 1)) + 1)) is complex real ext-real Element of REAL
(- 1) |^ (2 * (b + 1)) is complex real ext-real Element of REAL
((- 1) GeoSeq) . (2 * (b + 1)) is complex Element of COMPLEX
2 * (2 * (b + 1)) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * (2 * (b + 1))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
1 |^ ((2 * (2 * (b + 1))) + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 GeoSeq) . ((2 * (2 * (b + 1))) + 1) is complex Element of COMPLEX
((- 1) |^ (2 * (b + 1))) * (1 |^ ((2 * (2 * (b + 1))) + 1)) is complex real ext-real Element of REAL
((2 * (2 * (b + 1))) + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((2 * (2 * (b + 1))) + 1) is complex real ext-real Element of REAL
(((- 1) |^ (2 * (b + 1))) * (1 |^ ((2 * (2 * (b + 1))) + 1))) / (((2 * (2 * (b + 1))) + 1) !) is complex real ext-real Element of REAL
(((2 * (2 * (b + 1))) + 1) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ (2 * (b + 1))) * (1 |^ ((2 * (2 * (b + 1))) + 1))) * ((((2 * (2 * (b + 1))) + 1) !) ") is complex real ext-real set
1 * (1 |^ ((2 * (2 * (b + 1))) + 1)) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 * (1 |^ ((2 * (2 * (b + 1))) + 1))) / (((2 * (2 * (b + 1))) + 1) !) is complex real ext-real non negative Element of REAL
(1 * (1 |^ ((2 * (2 * (b + 1))) + 1))) * ((((2 * (2 * (b + 1))) + 1) !) ") is complex real ext-real non negative set
1 / (((2 * (2 * (b + 1))) + 1) !) is non empty complex real ext-real positive non negative Element of REAL
1 * ((((2 * (2 * (b + 1))) + 1) !) ") is non empty complex real ext-real positive non negative set
(- 1) |^ ((2 * (b + 1)) + 1) is complex real ext-real Element of REAL
((- 1) GeoSeq) . ((2 * (b + 1)) + 1) is complex Element of COMPLEX
2 * ((2 * (b + 1)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * ((2 * (b + 1)) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
1 |^ ((2 * ((2 * (b + 1)) + 1)) + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(1 GeoSeq) . ((2 * ((2 * (b + 1)) + 1)) + 1) is complex Element of COMPLEX
((- 1) |^ ((2 * (b + 1)) + 1)) * (1 |^ ((2 * ((2 * (b + 1)) + 1)) + 1)) is complex real ext-real Element of REAL
((2 * ((2 * (b + 1)) + 1)) + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((2 * ((2 * (b + 1)) + 1)) + 1) is complex real ext-real Element of REAL
(((- 1) |^ ((2 * (b + 1)) + 1)) * (1 |^ ((2 * ((2 * (b + 1)) + 1)) + 1))) / (((2 * ((2 * (b + 1)) + 1)) + 1) !) is complex real ext-real Element of REAL
(((2 * ((2 * (b + 1)) + 1)) + 1) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ ((2 * (b + 1)) + 1)) * (1 |^ ((2 * ((2 * (b + 1)) + 1)) + 1))) * ((((2 * ((2 * (b + 1)) + 1)) + 1) !) ") is complex real ext-real set
(- 1) * (1 |^ ((2 * ((2 * (b + 1)) + 1)) + 1)) is complex real ext-real non positive Element of REAL
((- 1) * (1 |^ ((2 * ((2 * (b + 1)) + 1)) + 1))) / (((2 * ((2 * (b + 1)) + 1)) + 1) !) is complex real ext-real non positive Element of REAL
((- 1) * (1 |^ ((2 * ((2 * (b + 1)) + 1)) + 1))) * ((((2 * ((2 * (b + 1)) + 1)) + 1) !) ") is complex real ext-real non positive set
((- 1) * 1) / (((2 * ((2 * (b + 1)) + 1)) + 1) !) is non empty complex real ext-real non positive negative Element of REAL
((- 1) * 1) * ((((2 * ((2 * (b + 1)) + 1)) + 1) !) ") is non empty complex real ext-real non positive negative set
(- 1) / (((2 * ((2 * (b + 1)) + 1)) + 1) !) is non empty complex real ext-real non positive negative Element of REAL
(- 1) * ((((2 * ((2 * (b + 1)) + 1)) + 1) !) ") is non empty complex real ext-real non positive negative set
1 / (((2 * ((2 * (b + 1)) + 1)) + 1) !) is non empty complex real ext-real positive non negative Element of REAL
1 * ((((2 * ((2 * (b + 1)) + 1)) + 1) !) ") is non empty complex real ext-real positive non negative set
(1 / (((2 * (2 * (b + 1))) + 1) !)) - (1 / (((2 * ((2 * (b + 1)) + 1)) + 1) !)) is complex real ext-real Element of REAL
- (1 / (((2 * ((2 * (b + 1)) + 1)) + 1) !)) is non empty complex real ext-real non positive negative set
(1 / (((2 * (2 * (b + 1))) + 1) !)) + (- (1 / (((2 * ((2 * (b + 1)) + 1)) + 1) !))) is complex real ext-real set
(() . 1) ^2 is complex real ext-real Element of REAL
(() . 1) * (() . 1) is complex real ext-real set
(() . 1) ^2 is complex real ext-real Element of REAL
(() . 1) * (() . 1) is complex real ext-real set
((() . 1) ^2) + ((() . 1) ^2) is complex real ext-real Element of REAL
(5 / 6) ^2 is complex real ext-real Element of REAL
(5 / 6) * (5 / 6) is non empty complex real ext-real positive non negative set
1 - ((() . 1) ^2) is complex real ext-real Element of REAL
- ((() . 1) ^2) is complex real ext-real set
1 + (- ((() . 1) ^2)) is complex real ext-real set
1 - (1 - ((() . 1) ^2)) is complex real ext-real Element of REAL
- (1 - ((() . 1) ^2)) is complex real ext-real set
1 + (- (1 - ((() . 1) ^2))) is complex real ext-real set
25 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
36 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
25 / 36 is non empty complex real ext-real positive non negative Element of REAL
36 " is non empty complex real ext-real positive non negative set
25 * (36 ") is non empty complex real ext-real positive non negative set
1 - (25 / 36) is complex real ext-real Element of REAL
- (25 / 36) is non empty complex real ext-real non positive negative set
1 + (- (25 / 36)) is complex real ext-real set
sqrt ((() . 1) ^2) is complex real ext-real Element of REAL
sqrt ((() . 1) ^2) is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Re (a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a) . b is complex real ext-real Element of REAL
(Re (a)) . b is complex real ext-real Element of REAL
(a) . b is complex Element of COMPLEX
Re ((a) . b) is complex real ext-real Element of REAL
a |^ b is complex real ext-real Element of REAL
a GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a GeoSeq) . b is complex Element of COMPLEX
b ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . b is complex real ext-real Element of REAL
(b !) + (0 * <i>) is complex set
(a |^ b) / ((b !) + (0 * <i>)) is complex Element of COMPLEX
((b !) + (0 * <i>)) " is complex set
(a |^ b) * (((b !) + (0 * <i>)) ") is complex set
Re ((a |^ b) / ((b !) + (0 * <i>))) is complex real ext-real Element of REAL
(a |^ b) / (b !) is complex real ext-real Element of REAL
(b !) " is non empty complex real ext-real positive non negative set
(a |^ b) * ((b !) ") is complex real ext-real set
((a |^ b) / (b !)) + (0 * <i>) is complex set
Re (((a |^ b) / (b !)) + (0 * <i>)) is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (a) is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
Re (Sum (a)) is complex real ext-real Element of REAL
Re (a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
Sum (Re (a)) is complex real ext-real Element of REAL
Im (a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
Sum (Im (a)) is complex real ext-real Element of REAL
(Sum (Im (a))) * <i> is complex set
(Sum (Re (a))) + ((Sum (Im (a))) * <i>) is complex set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b is complex set
(b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
(b) . 1 is complex Element of COMPLEX
(b) . 0 is complex Element of COMPLEX
b |^ a is complex set
b GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(b GeoSeq) . a is complex Element of COMPLEX
|.(b |^ a).| is complex real ext-real Element of REAL
Re (b |^ a) is complex real ext-real Element of REAL
(Re (b |^ a)) ^2 is complex real ext-real Element of REAL
(Re (b |^ a)) * (Re (b |^ a)) is complex real ext-real set
Im (b |^ a) is complex real ext-real Element of REAL
(Im (b |^ a)) ^2 is complex real ext-real Element of REAL
(Im (b |^ a)) * (Im (b |^ a)) is complex real ext-real set
((Re (b |^ a)) ^2) + ((Im (b |^ a)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (b |^ a)) ^2) + ((Im (b |^ a)) ^2)) is complex real ext-real Element of REAL
|.b.| is complex real ext-real Element of REAL
Re b is complex real ext-real Element of REAL
(Re b) ^2 is complex real ext-real Element of REAL
(Re b) * (Re b) is complex real ext-real set
Im b is complex real ext-real Element of REAL
(Im b) ^2 is complex real ext-real Element of REAL
(Im b) * (Im b) is complex real ext-real set
((Re b) ^2) + ((Im b) ^2) is complex real ext-real Element of REAL
sqrt (((Re b) ^2) + ((Im b) ^2)) is complex real ext-real Element of REAL
|.b.| |^ a is complex real ext-real Element of REAL
|.b.| GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(|.b.| GeoSeq) . a is complex Element of COMPLEX
b |^ 1 is complex set
(b GeoSeq) . 1 is complex Element of COMPLEX
b / (1 !) is complex Element of COMPLEX
(1 !) " is non empty complex real ext-real positive non negative set
b * ((1 !) ") is complex set
b |^ 0 is complex set
(b GeoSeq) . 0 is complex Element of COMPLEX
(b |^ 0) / (0 !) is complex Element of COMPLEX
(0 !) " is non empty complex real ext-real positive non negative set
(b |^ 0) * ((0 !) ") is complex set
|.(b |^ 0).| is complex real ext-real Element of REAL
Re (b |^ 0) is complex real ext-real Element of REAL
(Re (b |^ 0)) ^2 is complex real ext-real Element of REAL
(Re (b |^ 0)) * (Re (b |^ 0)) is complex real ext-real set
Im (b |^ 0) is complex real ext-real Element of REAL
(Im (b |^ 0)) ^2 is complex real ext-real Element of REAL
(Im (b |^ 0)) * (Im (b |^ 0)) is complex real ext-real set
((Re (b |^ 0)) ^2) + ((Im (b |^ 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (b |^ 0)) ^2) + ((Im (b |^ 0)) ^2)) is complex real ext-real Element of REAL
|.b.| |^ 0 is complex real ext-real Element of REAL
(|.b.| GeoSeq) . 0 is complex Element of COMPLEX
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b |^ th1 is complex set
(b GeoSeq) . th1 is complex Element of COMPLEX
|.(b |^ th1).| is complex real ext-real Element of REAL
Re (b |^ th1) is complex real ext-real Element of REAL
(Re (b |^ th1)) ^2 is complex real ext-real Element of REAL
(Re (b |^ th1)) * (Re (b |^ th1)) is complex real ext-real set
Im (b |^ th1) is complex real ext-real Element of REAL
(Im (b |^ th1)) ^2 is complex real ext-real Element of REAL
(Im (b |^ th1)) * (Im (b |^ th1)) is complex real ext-real set
((Re (b |^ th1)) ^2) + ((Im (b |^ th1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (b |^ th1)) ^2) + ((Im (b |^ th1)) ^2)) is complex real ext-real Element of REAL
|.b.| |^ th1 is complex real ext-real Element of REAL
(|.b.| GeoSeq) . th1 is complex Element of COMPLEX
th1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b |^ (th1 + 1) is complex set
(b GeoSeq) . (th1 + 1) is complex Element of COMPLEX
|.(b |^ (th1 + 1)).| is complex real ext-real Element of REAL
Re (b |^ (th1 + 1)) is complex real ext-real Element of REAL
(Re (b |^ (th1 + 1))) ^2 is complex real ext-real Element of REAL
(Re (b |^ (th1 + 1))) * (Re (b |^ (th1 + 1))) is complex real ext-real set
Im (b |^ (th1 + 1)) is complex real ext-real Element of REAL
(Im (b |^ (th1 + 1))) ^2 is complex real ext-real Element of REAL
(Im (b |^ (th1 + 1))) * (Im (b |^ (th1 + 1))) is complex real ext-real set
((Re (b |^ (th1 + 1))) ^2) + ((Im (b |^ (th1 + 1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (b |^ (th1 + 1))) ^2) + ((Im (b |^ (th1 + 1))) ^2)) is complex real ext-real Element of REAL
|.b.| |^ (th1 + 1) is complex real ext-real Element of REAL
(|.b.| GeoSeq) . (th1 + 1) is complex Element of COMPLEX
b * ((b GeoSeq) . th1) is complex set
|.(b * ((b GeoSeq) . th1)).| is complex real ext-real Element of REAL
Re (b * ((b GeoSeq) . th1)) is complex real ext-real Element of REAL
(Re (b * ((b GeoSeq) . th1))) ^2 is complex real ext-real Element of REAL
(Re (b * ((b GeoSeq) . th1))) * (Re (b * ((b GeoSeq) . th1))) is complex real ext-real set
Im (b * ((b GeoSeq) . th1)) is complex real ext-real Element of REAL
(Im (b * ((b GeoSeq) . th1))) ^2 is complex real ext-real Element of REAL
(Im (b * ((b GeoSeq) . th1))) * (Im (b * ((b GeoSeq) . th1))) is complex real ext-real set
((Re (b * ((b GeoSeq) . th1))) ^2) + ((Im (b * ((b GeoSeq) . th1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (b * ((b GeoSeq) . th1))) ^2) + ((Im (b * ((b GeoSeq) . th1))) ^2)) is complex real ext-real Element of REAL
(|.b.| |^ th1) * |.b.| is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
(a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (a) is complex real ext-real Element of REAL
Im (Partial_Sums (a)) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent bounded_above bounded_below Element of K19(K20(NAT,REAL))
(Im (Partial_Sums (a))) . 0 is complex real ext-real Element of REAL
(Partial_Sums (a)) . 0 is complex Element of COMPLEX
Im ((Partial_Sums (a)) . 0) is complex real ext-real Element of REAL
(a) . 0 is complex Element of COMPLEX
Im ((a) . 0) is complex real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative set
(Im (Partial_Sums (a))) . b is complex real ext-real Element of REAL
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Im (Partial_Sums (a))) . (b + 1) is complex real ext-real Element of REAL
(Partial_Sums (a)) . (b + 1) is complex Element of COMPLEX
Im ((Partial_Sums (a)) . (b + 1)) is complex real ext-real Element of REAL
(Partial_Sums (a)) . b is complex Element of COMPLEX
(a) . (b + 1) is complex Element of COMPLEX
((Partial_Sums (a)) . b) + ((a) . (b + 1)) is complex Element of COMPLEX
Im (((Partial_Sums (a)) . b) + ((a) . (b + 1))) is complex real ext-real Element of REAL
Im ((Partial_Sums (a)) . b) is complex real ext-real Element of REAL
Im ((a) . (b + 1)) is complex real ext-real Element of REAL
(Im ((Partial_Sums (a)) . b)) + (Im ((a) . (b + 1))) is complex real ext-real Element of REAL
0 + (Im ((a) . (b + 1))) is complex real ext-real Element of REAL
a |^ (b + 1) is complex real ext-real Element of REAL
a GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a GeoSeq) . (b + 1) is complex Element of COMPLEX
(b + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (b + 1) is complex real ext-real Element of REAL
(a |^ (b + 1)) / ((b + 1) !) is complex real ext-real Element of REAL
((b + 1) !) " is non empty complex real ext-real positive non negative set
(a |^ (b + 1)) * (((b + 1) !) ") is complex real ext-real set
((a |^ (b + 1)) / ((b + 1) !)) + (0 * <i>) is complex set
Im (((a |^ (b + 1)) / ((b + 1) !)) + (0 * <i>)) is complex real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative set
(Im (Partial_Sums (a))) . b is complex real ext-real Element of REAL
lim (Im (Partial_Sums (a))) is complex real ext-real Element of REAL
(Im (Partial_Sums (a))) . 0 is complex real ext-real Element of REAL
Im (Sum (a)) is complex real ext-real Element of REAL
Re (Sum (a)) is complex real ext-real Element of REAL
(Re (Sum (a))) + (0 * <i>) is complex set
a is complex real ext-real set
b is complex real ext-real set
a + b is complex real ext-real set
((a + b)) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum ((a + b)) is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (a) is complex real ext-real Element of REAL
(b) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (b) is complex real ext-real Element of REAL
(Sum (a)) * (Sum (b)) is complex real ext-real Element of REAL
th1 is complex real ext-real Element of REAL
th2 is complex real ext-real Element of REAL
th1 + th2 is complex real ext-real Element of REAL
((th1 + th2)) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum ((th1 + th2)) is complex real ext-real Element of REAL
((th1 + th2)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((th1 + th2)) is complex Element of COMPLEX
Partial_Sums ((th1 + th2)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((th1 + th2))) is complex Element of COMPLEX
Re (Sum ((th1 + th2))) is complex real ext-real Element of REAL
(th1) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (th1) is complex Element of COMPLEX
Partial_Sums (th1) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (th1)) is complex Element of COMPLEX
(th2) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (th2) is complex Element of COMPLEX
Partial_Sums (th2) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (th2)) is complex Element of COMPLEX
(Sum (th1)) * (Sum (th2)) is complex Element of COMPLEX
Re ((Sum (th1)) * (Sum (th2))) is complex real ext-real Element of REAL
Re (Sum (th1)) is complex real ext-real Element of REAL
Im (Sum (th1)) is complex real ext-real Element of REAL
(Im (Sum (th1))) * <i> is complex set
(Re (Sum (th1))) + ((Im (Sum (th1))) * <i>) is complex set
((Re (Sum (th1))) + ((Im (Sum (th1))) * <i>)) * (Sum (th2)) is complex set
Re (((Re (Sum (th1))) + ((Im (Sum (th1))) * <i>)) * (Sum (th2))) is complex real ext-real Element of REAL
Re (Sum (th2)) is complex real ext-real Element of REAL
Im (Sum (th2)) is complex real ext-real Element of REAL
(Im (Sum (th2))) * <i> is complex set
(Re (Sum (th2))) + ((Im (Sum (th2))) * <i>) is complex set
((Re (Sum (th1))) + ((Im (Sum (th1))) * <i>)) * ((Re (Sum (th2))) + ((Im (Sum (th2))) * <i>)) is complex set
Re (((Re (Sum (th1))) + ((Im (Sum (th1))) * <i>)) * ((Re (Sum (th2))) + ((Im (Sum (th2))) * <i>))) is complex real ext-real Element of REAL
(th1) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (th1) is complex real ext-real Element of REAL
(Sum (th1)) + ((Im (Sum (th1))) * <i>) is complex set
((Sum (th1)) + ((Im (Sum (th1))) * <i>)) * ((Re (Sum (th2))) + ((Im (Sum (th2))) * <i>)) is complex set
Re (((Sum (th1)) + ((Im (Sum (th1))) * <i>)) * ((Re (Sum (th2))) + ((Im (Sum (th2))) * <i>))) is complex real ext-real Element of REAL
(Sum (th1)) + (0 * <i>) is complex set
((Sum (th1)) + (0 * <i>)) * ((Re (Sum (th2))) + ((Im (Sum (th2))) * <i>)) is complex set
Re (((Sum (th1)) + (0 * <i>)) * ((Re (Sum (th2))) + ((Im (Sum (th2))) * <i>))) is complex real ext-real Element of REAL
(th2) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (th2) is complex real ext-real Element of REAL
(Sum (th2)) + ((Im (Sum (th2))) * <i>) is complex set
(Sum (th1)) * ((Sum (th2)) + ((Im (Sum (th2))) * <i>)) is complex set
Re ((Sum (th1)) * ((Sum (th2)) + ((Im (Sum (th2))) * <i>))) is complex real ext-real Element of REAL
(Sum (th2)) + (0 * <i>) is complex set
(Sum (th1)) * ((Sum (th2)) + (0 * <i>)) is complex set
Re ((Sum (th1)) * ((Sum (th2)) + (0 * <i>))) is complex real ext-real Element of REAL
(Sum (th1)) * (Sum (th2)) is complex real ext-real Element of REAL
((Sum (th1)) * (Sum (th2))) + (0 * <i>) is complex set
Re (((Sum (th1)) * (Sum (th2))) + (0 * <i>)) is complex real ext-real Element of REAL
a is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
b is complex real ext-real set
a . b is complex real ext-real Element of REAL
(b) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (b) is complex real ext-real Element of REAL
a is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
b is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
th1 is complex real ext-real Element of REAL
a . th1 is complex real ext-real Element of REAL
b . th1 is complex real ext-real Element of REAL
(th1) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (th1) is complex real ext-real Element of REAL
() is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
a is complex real ext-real set
() . a is complex real ext-real Element of REAL
a is complex real ext-real set
(a) is set
() . a is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
dom () is V46() V47() V48() Element of K19(REAL)
a is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
Re (Sum (a)) is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (a) is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
(a) is complex Element of COMPLEX
() . a is complex set
(a) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
(a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (a) is complex real ext-real Element of REAL
a is complex real ext-real set
b is complex real ext-real set
a + b is complex real ext-real set
() . (a + b) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
() . b is complex real ext-real Element of REAL
(() . a) * (() . b) is complex real ext-real Element of REAL
((a + b)) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum ((a + b)) is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (a) is complex real ext-real Element of REAL
(b) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (b) is complex real ext-real Element of REAL
(Sum (a)) * (Sum (b)) is complex real ext-real Element of REAL
(() . a) * (Sum (b)) is complex real ext-real Element of REAL
a is complex real ext-real set
b is complex real ext-real set
a + b is complex real ext-real set
((a + b)) is complex real ext-real set
() . (a + b) is complex real ext-real Element of REAL
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
(b) is complex real ext-real set
() . b is complex real ext-real Element of REAL
(a) * (b) is complex real ext-real set
() . 0 is complex real ext-real Element of REAL
(0) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (0) is complex real ext-real Element of REAL
(0) is complex real ext-real Element of REAL
a is complex real ext-real set
() . a is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(Partial_Sums (a)) . 0 is complex real ext-real Element of REAL
(a) . 0 is complex real ext-real Element of REAL
a |^ 0 is complex real ext-real set
a GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a GeoSeq) . 0 is complex Element of COMPLEX
(a |^ 0) / (0 !) is complex real ext-real Element of REAL
(0 !) " is non empty complex real ext-real positive non negative set
(a |^ 0) * ((0 !) ") is complex real ext-real set
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (a)) . b is complex real ext-real Element of REAL
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (a)) . (b + 1) is complex real ext-real Element of REAL
(a) . (b + 1) is complex real ext-real Element of REAL
((Partial_Sums (a)) . b) + ((a) . (b + 1)) is complex real ext-real Element of REAL
a |^ (b + 1) is complex real ext-real set
(a GeoSeq) . (b + 1) is complex Element of COMPLEX
(b + 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (b + 1) is complex real ext-real Element of REAL
(a |^ (b + 1)) / ((b + 1) !) is complex real ext-real Element of REAL
((b + 1) !) " is non empty complex real ext-real positive non negative set
(a |^ (b + 1)) * (((b + 1) !) ") is complex real ext-real set
((Partial_Sums (a)) . b) + ((a |^ (b + 1)) / ((b + 1) !)) is complex real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (a)) . b is complex real ext-real Element of REAL
lim (Partial_Sums (a)) is complex real ext-real Element of REAL
Sum (a) is complex real ext-real Element of REAL
a is complex real ext-real set
() . a is complex real ext-real Element of REAL
- a is complex real ext-real set
() . (- a) is complex real ext-real Element of REAL
(() . (- a)) * (() . a) is complex real ext-real Element of REAL
(- a) + a is complex real ext-real set
() . ((- a) + a) is complex real ext-real Element of REAL
1 / (() . (- a)) is complex real ext-real Element of REAL
(() . (- a)) " is complex real ext-real set
1 * ((() . (- a)) ") is complex real ext-real set
a is complex real ext-real set
() . a is complex real ext-real Element of REAL
a is complex real ext-real set
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
a is complex set
a is complex real ext-real Element of REAL
a * <i> is complex set
b is complex set
(a * <i>) * b is complex set
Re ((a * <i>) * b) is complex real ext-real Element of REAL
Im b is complex real ext-real Element of REAL
a * (Im b) is complex real ext-real Element of REAL
- (a * (Im b)) is complex real ext-real Element of REAL
Im ((a * <i>) * b) is complex real ext-real Element of REAL
Re b is complex real ext-real Element of REAL
a * (Re b) is complex real ext-real Element of REAL
a * b is complex set
Re (a * b) is complex real ext-real Element of REAL
Im (a * b) is complex real ext-real Element of REAL
(Im b) * <i> is complex set
(Re b) + ((Im b) * <i>) is complex set
(a * <i>) * ((Re b) + ((Im b) * <i>)) is complex set
(a * (Re b)) * <i> is complex set
(- (a * (Im b))) + ((a * (Re b)) * <i>) is complex set
a * ((Re b) + ((Im b) * <i>)) is complex set
(a * (Im b)) * <i> is complex set
(a * (Re b)) + ((a * (Im b)) * <i>) is complex set
a is complex real ext-real set
a * <i> is complex set
b is complex set
b / (a * <i>) is complex Element of COMPLEX
(a * <i>) " is complex set
b * ((a * <i>) ") is complex set
Re (b / (a * <i>)) is complex real ext-real Element of REAL
Im b is complex real ext-real Element of REAL
(Im b) / a is complex real ext-real Element of REAL
a " is complex real ext-real set
(Im b) * (a ") is complex real ext-real set
Im (b / (a * <i>)) is complex real ext-real Element of REAL
Re b is complex real ext-real Element of REAL
(Re b) / a is complex real ext-real Element of REAL
(Re b) * (a ") is complex real ext-real set
- ((Re b) / a) is complex real ext-real Element of REAL
b / a is complex Element of COMPLEX
b * (a ") is complex set
|.(b / a).| is complex real ext-real Element of REAL
Re (b / a) is complex real ext-real Element of REAL
(Re (b / a)) ^2 is complex real ext-real Element of REAL
(Re (b / a)) * (Re (b / a)) is complex real ext-real set
Im (b / a) is complex real ext-real Element of REAL
(Im (b / a)) ^2 is complex real ext-real Element of REAL
(Im (b / a)) * (Im (b / a)) is complex real ext-real set
((Re (b / a)) ^2) + ((Im (b / a)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (b / a)) ^2) + ((Im (b / a)) ^2)) is complex real ext-real Element of REAL
|.b.| is complex real ext-real Element of REAL
(Re b) ^2 is complex real ext-real Element of REAL
(Re b) * (Re b) is complex real ext-real set
(Im b) ^2 is complex real ext-real Element of REAL
(Im b) * (Im b) is complex real ext-real set
((Re b) ^2) + ((Im b) ^2) is complex real ext-real Element of REAL
sqrt (((Re b) ^2) + ((Im b) ^2)) is complex real ext-real Element of REAL
|.b.| / a is complex real ext-real Element of REAL
|.b.| * (a ") is complex real ext-real set
0 + (a * <i>) is complex set
Re (0 + (a * <i>)) is complex real ext-real Element of REAL
Im (0 + (a * <i>)) is complex real ext-real Element of REAL
a + (0 * <i>) is complex set
Im (a + (0 * <i>)) is complex real ext-real Element of REAL
(Re b) * 0 is complex real ext-real Element of REAL
(Im b) * a is complex real ext-real Element of REAL
((Re b) * 0) + ((Im b) * a) is complex real ext-real Element of REAL
0 ^2 is complex real ext-real Element of REAL
0 * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
a ^2 is complex real ext-real set
a * a is complex real ext-real set
(0 ^2) + (a ^2) is complex real ext-real Element of REAL
(((Re b) * 0) + ((Im b) * a)) / ((0 ^2) + (a ^2)) is complex real ext-real Element of REAL
((0 ^2) + (a ^2)) " is complex real ext-real set
(((Re b) * 0) + ((Im b) * a)) * (((0 ^2) + (a ^2)) ") is complex real ext-real set
0 * (Im b) is complex real ext-real Element of REAL
(Re b) * a is complex real ext-real Element of REAL
(0 * (Im b)) - ((Re b) * a) is complex real ext-real Element of REAL
- ((Re b) * a) is complex real ext-real set
(0 * (Im b)) + (- ((Re b) * a)) is complex real ext-real set
((0 * (Im b)) - ((Re b) * a)) / ((0 ^2) + (a ^2)) is complex real ext-real Element of REAL
((0 * (Im b)) - ((Re b) * a)) * (((0 ^2) + (a ^2)) ") is complex real ext-real set
(((0 * (Im b)) - ((Re b) * a)) / ((0 ^2) + (a ^2))) * <i> is complex set
((((Re b) * 0) + ((Im b) * a)) / ((0 ^2) + (a ^2))) + ((((0 * (Im b)) - ((Re b) * a)) / ((0 ^2) + (a ^2))) * <i>) is complex set
((Im b) * a) / (a * a) is complex real ext-real Element of REAL
(a * a) " is complex real ext-real set
((Im b) * a) * ((a * a) ") is complex real ext-real set
- (Re b) is complex real ext-real Element of REAL
(- (Re b)) * a is complex real ext-real Element of REAL
((- (Re b)) * a) / (a * a) is complex real ext-real Element of REAL
((- (Re b)) * a) * ((a * a) ") is complex real ext-real set
(((- (Re b)) * a) / (a * a)) * <i> is complex set
(((Im b) * a) / (a * a)) + ((((- (Re b)) * a) / (a * a)) * <i>) is complex set
(- (Re b)) / a is complex real ext-real Element of REAL
(- (Re b)) * (a ") is complex real ext-real set
((- (Re b)) / a) * <i> is complex set
(((Im b) * a) / (a * a)) + (((- (Re b)) / a) * <i>) is complex set
(- ((Re b) / a)) * <i> is complex set
((Im b) / a) + ((- ((Re b) / a)) * <i>) is complex set
|.a.| is complex real ext-real Element of REAL
Re a is complex real ext-real Element of REAL
(Re a) ^2 is complex real ext-real Element of REAL
(Re a) * (Re a) is complex real ext-real set
Im a is complex real ext-real Element of REAL
(Im a) ^2 is complex real ext-real Element of REAL
(Im a) * (Im a) is complex real ext-real set
((Re a) ^2) + ((Im a) ^2) is complex real ext-real Element of REAL
sqrt (((Re a) ^2) + ((Im a) ^2)) is complex real ext-real Element of REAL
|.b.| / |.a.| is complex real ext-real Element of REAL
|.a.| " is complex real ext-real set
|.b.| * (|.a.| ") is complex real ext-real set
sqrt (a ^2) is complex real ext-real set
|.b.| / (sqrt (a ^2)) is complex real ext-real Element of REAL
(sqrt (a ^2)) " is complex real ext-real set
|.b.| * ((sqrt (a ^2)) ") is complex real ext-real set
a is complex set
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
|.(a).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums |.(a).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued non-decreasing Element of K19(K20(NAT,REAL))
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
|.(a).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent summable bounded_above bounded_below Element of K19(K20(NAT,REAL))
Partial_Sums |.(a).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued non-decreasing bounded convergent bounded_above bounded_below Element of K19(K20(NAT,REAL))
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums |.(a).|) . (b + 1) is complex real ext-real Element of REAL
(Partial_Sums |.(a).|) . b is complex real ext-real Element of REAL
(Partial_Sums |.(a).|) . 0 is complex real ext-real Element of REAL
|.(a).| . 0 is complex real ext-real Element of REAL
(a) . 0 is complex Element of COMPLEX
|.((a) . 0).| is complex real ext-real Element of REAL
Re ((a) . 0) is complex real ext-real Element of REAL
(Re ((a) . 0)) ^2 is complex real ext-real Element of REAL
(Re ((a) . 0)) * (Re ((a) . 0)) is complex real ext-real set
Im ((a) . 0) is complex real ext-real Element of REAL
(Im ((a) . 0)) ^2 is complex real ext-real Element of REAL
(Im ((a) . 0)) * (Im ((a) . 0)) is complex real ext-real set
((Re ((a) . 0)) ^2) + ((Im ((a) . 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a) . 0)) ^2) + ((Im ((a) . 0)) ^2)) is complex real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a |^ (0 + 1) is complex set
a GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a GeoSeq) . (0 + 1) is complex Element of COMPLEX
0 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(0 + 2) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (0 + 2) is complex real ext-real Element of REAL
(a |^ (0 + 1)) / ((0 + 2) !) is complex Element of COMPLEX
((0 + 2) !) " is non empty complex real ext-real positive non negative set
(a |^ (0 + 1)) * (((0 + 2) !) ") is complex set
|.((a |^ (0 + 1)) / ((0 + 2) !)).| is complex real ext-real Element of REAL
Re ((a |^ (0 + 1)) / ((0 + 2) !)) is complex real ext-real Element of REAL
(Re ((a |^ (0 + 1)) / ((0 + 2) !))) ^2 is complex real ext-real Element of REAL
(Re ((a |^ (0 + 1)) / ((0 + 2) !))) * (Re ((a |^ (0 + 1)) / ((0 + 2) !))) is complex real ext-real set
Im ((a |^ (0 + 1)) / ((0 + 2) !)) is complex real ext-real Element of REAL
(Im ((a |^ (0 + 1)) / ((0 + 2) !))) ^2 is complex real ext-real Element of REAL
(Im ((a |^ (0 + 1)) / ((0 + 2) !))) * (Im ((a |^ (0 + 1)) / ((0 + 2) !))) is complex real ext-real set
((Re ((a |^ (0 + 1)) / ((0 + 2) !))) ^2) + ((Im ((a |^ (0 + 1)) / ((0 + 2) !))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a |^ (0 + 1)) / ((0 + 2) !))) ^2) + ((Im ((a |^ (0 + 1)) / ((0 + 2) !))) ^2)) is complex real ext-real Element of REAL
a / 2 is complex Element of COMPLEX
2 " is non empty complex real ext-real positive non negative set
a * (2 ") is complex set
|.(a / 2).| is complex real ext-real Element of REAL
Re (a / 2) is complex real ext-real Element of REAL
(Re (a / 2)) ^2 is complex real ext-real Element of REAL
(Re (a / 2)) * (Re (a / 2)) is complex real ext-real set
Im (a / 2) is complex real ext-real Element of REAL
(Im (a / 2)) ^2 is complex real ext-real Element of REAL
(Im (a / 2)) * (Im (a / 2)) is complex real ext-real set
((Re (a / 2)) ^2) + ((Im (a / 2)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (a / 2)) ^2) + ((Im (a / 2)) ^2)) is complex real ext-real Element of REAL
|.a.| is complex real ext-real Element of REAL
Re a is complex real ext-real Element of REAL
(Re a) ^2 is complex real ext-real Element of REAL
(Re a) * (Re a) is complex real ext-real set
Im a is complex real ext-real Element of REAL
(Im a) ^2 is complex real ext-real Element of REAL
(Im a) * (Im a) is complex real ext-real set
((Re a) ^2) + ((Im a) ^2) is complex real ext-real Element of REAL
sqrt (((Re a) ^2) + ((Im a) ^2)) is complex real ext-real Element of REAL
|.a.| / 2 is complex real ext-real Element of REAL
|.a.| * (2 ") is complex real ext-real set
(Partial_Sums |.(a).|) . (0 + 1) is complex real ext-real Element of REAL
(Partial_Sums |.(a).|) . 0 is complex real ext-real Element of REAL
|.(a).| . (0 + 1) is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . 0) + (|.(a).| . (0 + 1)) is complex real ext-real Element of REAL
|.(a).| . 0 is complex real ext-real Element of REAL
|.(a).| . 1 is complex real ext-real Element of REAL
(|.(a).| . 0) + (|.(a).| . 1) is complex real ext-real Element of REAL
(a) . 1 is complex Element of COMPLEX
|.((a) . 1).| is complex real ext-real Element of REAL
Re ((a) . 1) is complex real ext-real Element of REAL
(Re ((a) . 1)) ^2 is complex real ext-real Element of REAL
(Re ((a) . 1)) * (Re ((a) . 1)) is complex real ext-real set
Im ((a) . 1) is complex real ext-real Element of REAL
(Im ((a) . 1)) ^2 is complex real ext-real Element of REAL
(Im ((a) . 1)) * (Im ((a) . 1)) is complex real ext-real set
((Re ((a) . 1)) ^2) + ((Im ((a) . 1)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a) . 1)) ^2) + ((Im ((a) . 1)) ^2)) is complex real ext-real Element of REAL
(|.(a).| . 0) + |.((a) . 1).| is complex real ext-real Element of REAL
(a) . 0 is complex Element of COMPLEX
|.((a) . 0).| is complex real ext-real Element of REAL
Re ((a) . 0) is complex real ext-real Element of REAL
(Re ((a) . 0)) ^2 is complex real ext-real Element of REAL
(Re ((a) . 0)) * (Re ((a) . 0)) is complex real ext-real set
Im ((a) . 0) is complex real ext-real Element of REAL
(Im ((a) . 0)) ^2 is complex real ext-real Element of REAL
(Im ((a) . 0)) * (Im ((a) . 0)) is complex real ext-real set
((Re ((a) . 0)) ^2) + ((Im ((a) . 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a) . 0)) ^2) + ((Im ((a) . 0)) ^2)) is complex real ext-real Element of REAL
|.((a) . 0).| + |.((a) . 1).| is complex real ext-real Element of REAL
|.((a) . 0).| + |.a.| is complex real ext-real Element of REAL
1 + |.a.| is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . (0 + 1)) - ((Partial_Sums |.(a).|) . 0) is complex real ext-real Element of REAL
- ((Partial_Sums |.(a).|) . 0) is complex real ext-real set
((Partial_Sums |.(a).|) . (0 + 1)) + (- ((Partial_Sums |.(a).|) . 0)) is complex real ext-real set
|.a.| - (|.a.| / 2) is complex real ext-real Element of REAL
- (|.a.| / 2) is complex real ext-real set
|.a.| + (- (|.a.| / 2)) is complex real ext-real set
1 + (|.a.| - (|.a.| / 2)) is complex real ext-real Element of REAL
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums |.(a).|) . (th1 + 1) is complex real ext-real Element of REAL
(Partial_Sums |.(a).|) . th1 is complex real ext-real Element of REAL
(th1 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums |.(a).|) . ((th1 + 1) + 1) is complex real ext-real Element of REAL
(Partial_Sums |.(a).|) . (th1 + 1) is complex real ext-real Element of REAL
|.(a).| . (th1 + 1) is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . th1) + (|.(a).| . (th1 + 1)) is complex real ext-real Element of REAL
(a) . (th1 + 1) is complex Element of COMPLEX
|.((a) . (th1 + 1)).| is complex real ext-real Element of REAL
Re ((a) . (th1 + 1)) is complex real ext-real Element of REAL
(Re ((a) . (th1 + 1))) ^2 is complex real ext-real Element of REAL
(Re ((a) . (th1 + 1))) * (Re ((a) . (th1 + 1))) is complex real ext-real set
Im ((a) . (th1 + 1)) is complex real ext-real Element of REAL
(Im ((a) . (th1 + 1))) ^2 is complex real ext-real Element of REAL
(Im ((a) . (th1 + 1))) * (Im ((a) . (th1 + 1))) is complex real ext-real set
((Re ((a) . (th1 + 1))) ^2) + ((Im ((a) . (th1 + 1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a) . (th1 + 1))) ^2) + ((Im ((a) . (th1 + 1))) ^2)) is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . th1) + |.((a) . (th1 + 1)).| is complex real ext-real Element of REAL
a |^ ((th1 + 1) + 1) is complex set
(a GeoSeq) . ((th1 + 1) + 1) is complex Element of COMPLEX
(th1 + 1) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((th1 + 1) + 2) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . ((th1 + 1) + 2) is complex real ext-real Element of REAL
(a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !) is complex Element of COMPLEX
(((th1 + 1) + 2) !) " is non empty complex real ext-real positive non negative set
(a |^ ((th1 + 1) + 1)) * ((((th1 + 1) + 2) !) ") is complex set
|.((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !)).| is complex real ext-real Element of REAL
Re ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !)) is complex real ext-real Element of REAL
(Re ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !))) ^2 is complex real ext-real Element of REAL
(Re ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !))) * (Re ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !))) is complex real ext-real set
Im ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !)) is complex real ext-real Element of REAL
(Im ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !))) ^2 is complex real ext-real Element of REAL
(Im ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !))) * (Im ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !))) is complex real ext-real set
((Re ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !))) ^2) + ((Im ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !))) ^2) + ((Im ((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !))) ^2)) is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . th1) + |.((a |^ ((th1 + 1) + 1)) / (((th1 + 1) + 2) !)).| is complex real ext-real Element of REAL
th1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a |^ (th1 + 2) is complex set
(a GeoSeq) . (th1 + 2) is complex Element of COMPLEX
3 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 + 3 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(th1 + 3) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (th1 + 3) is complex real ext-real Element of REAL
(a |^ (th1 + 2)) / ((th1 + 3) !) is complex Element of COMPLEX
((th1 + 3) !) " is non empty complex real ext-real positive non negative set
(a |^ (th1 + 2)) * (((th1 + 3) !) ") is complex set
|.((a |^ (th1 + 2)) / ((th1 + 3) !)).| is complex real ext-real Element of REAL
Re ((a |^ (th1 + 2)) / ((th1 + 3) !)) is complex real ext-real Element of REAL
(Re ((a |^ (th1 + 2)) / ((th1 + 3) !))) ^2 is complex real ext-real Element of REAL
(Re ((a |^ (th1 + 2)) / ((th1 + 3) !))) * (Re ((a |^ (th1 + 2)) / ((th1 + 3) !))) is complex real ext-real set
Im ((a |^ (th1 + 2)) / ((th1 + 3) !)) is complex real ext-real Element of REAL
(Im ((a |^ (th1 + 2)) / ((th1 + 3) !))) ^2 is complex real ext-real Element of REAL
(Im ((a |^ (th1 + 2)) / ((th1 + 3) !))) * (Im ((a |^ (th1 + 2)) / ((th1 + 3) !))) is complex real ext-real set
((Re ((a |^ (th1 + 2)) / ((th1 + 3) !))) ^2) + ((Im ((a |^ (th1 + 2)) / ((th1 + 3) !))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a |^ (th1 + 2)) / ((th1 + 3) !))) ^2) + ((Im ((a |^ (th1 + 2)) / ((th1 + 3) !))) ^2)) is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . th1) + |.((a |^ (th1 + 2)) / ((th1 + 3) !)).| is complex real ext-real Element of REAL
a #N (th1 + 2) is complex Element of COMPLEX
|.(a #N (th1 + 2)).| is complex real ext-real Element of REAL
Re (a #N (th1 + 2)) is complex real ext-real Element of REAL
(Re (a #N (th1 + 2))) ^2 is complex real ext-real Element of REAL
(Re (a #N (th1 + 2))) * (Re (a #N (th1 + 2))) is complex real ext-real set
Im (a #N (th1 + 2)) is complex real ext-real Element of REAL
(Im (a #N (th1 + 2))) ^2 is complex real ext-real Element of REAL
(Im (a #N (th1 + 2))) * (Im (a #N (th1 + 2))) is complex real ext-real set
((Re (a #N (th1 + 2))) ^2) + ((Im (a #N (th1 + 2))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (a #N (th1 + 2))) ^2) + ((Im (a #N (th1 + 2))) ^2)) is complex real ext-real Element of REAL
|.(a #N (th1 + 2)).| / ((th1 + 3) !) is complex real ext-real Element of REAL
|.(a #N (th1 + 2)).| * (((th1 + 3) !) ") is complex real ext-real set
((Partial_Sums |.(a).|) . th1) + (|.(a #N (th1 + 2)).| / ((th1 + 3) !)) is complex real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th1 + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
|.(a).| . (th1 + (1 + 1)) is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . (th1 + 1)) + (|.(a).| . (th1 + (1 + 1))) is complex real ext-real Element of REAL
(a) . (th1 + 2) is complex Element of COMPLEX
|.((a) . (th1 + 2)).| is complex real ext-real Element of REAL
Re ((a) . (th1 + 2)) is complex real ext-real Element of REAL
(Re ((a) . (th1 + 2))) ^2 is complex real ext-real Element of REAL
(Re ((a) . (th1 + 2))) * (Re ((a) . (th1 + 2))) is complex real ext-real set
Im ((a) . (th1 + 2)) is complex real ext-real Element of REAL
(Im ((a) . (th1 + 2))) ^2 is complex real ext-real Element of REAL
(Im ((a) . (th1 + 2))) * (Im ((a) . (th1 + 2))) is complex real ext-real set
((Re ((a) . (th1 + 2))) ^2) + ((Im ((a) . (th1 + 2))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a) . (th1 + 2))) ^2) + ((Im ((a) . (th1 + 2))) ^2)) is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . (th1 + 1)) + |.((a) . (th1 + 2)).| is complex real ext-real Element of REAL
(th1 + 2) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (th1 + 2) is complex real ext-real Element of REAL
(a |^ (th1 + 2)) / ((th1 + 2) !) is complex Element of COMPLEX
((th1 + 2) !) " is non empty complex real ext-real positive non negative set
(a |^ (th1 + 2)) * (((th1 + 2) !) ") is complex set
|.((a |^ (th1 + 2)) / ((th1 + 2) !)).| is complex real ext-real Element of REAL
Re ((a |^ (th1 + 2)) / ((th1 + 2) !)) is complex real ext-real Element of REAL
(Re ((a |^ (th1 + 2)) / ((th1 + 2) !))) ^2 is complex real ext-real Element of REAL
(Re ((a |^ (th1 + 2)) / ((th1 + 2) !))) * (Re ((a |^ (th1 + 2)) / ((th1 + 2) !))) is complex real ext-real set
Im ((a |^ (th1 + 2)) / ((th1 + 2) !)) is complex real ext-real Element of REAL
(Im ((a |^ (th1 + 2)) / ((th1 + 2) !))) ^2 is complex real ext-real Element of REAL
(Im ((a |^ (th1 + 2)) / ((th1 + 2) !))) * (Im ((a |^ (th1 + 2)) / ((th1 + 2) !))) is complex real ext-real set
((Re ((a |^ (th1 + 2)) / ((th1 + 2) !))) ^2) + ((Im ((a |^ (th1 + 2)) / ((th1 + 2) !))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a |^ (th1 + 2)) / ((th1 + 2) !))) ^2) + ((Im ((a |^ (th1 + 2)) / ((th1 + 2) !))) ^2)) is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . (th1 + 1)) + |.((a |^ (th1 + 2)) / ((th1 + 2) !)).| is complex real ext-real Element of REAL
|.(a #N (th1 + 2)).| / ((th1 + 2) !) is complex real ext-real Element of REAL
|.(a #N (th1 + 2)).| * (((th1 + 2) !) ") is complex real ext-real set
((Partial_Sums |.(a).|) . (th1 + 1)) + (|.(a #N (th1 + 2)).| / ((th1 + 2) !)) is complex real ext-real Element of REAL
|.(a |^ (th1 + 2)).| is complex real ext-real Element of REAL
Re (a |^ (th1 + 2)) is complex real ext-real Element of REAL
(Re (a |^ (th1 + 2))) ^2 is complex real ext-real Element of REAL
(Re (a |^ (th1 + 2))) * (Re (a |^ (th1 + 2))) is complex real ext-real set
Im (a |^ (th1 + 2)) is complex real ext-real Element of REAL
(Im (a |^ (th1 + 2))) ^2 is complex real ext-real Element of REAL
(Im (a |^ (th1 + 2))) * (Im (a |^ (th1 + 2))) is complex real ext-real set
((Re (a |^ (th1 + 2))) ^2) + ((Im (a |^ (th1 + 2))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (a |^ (th1 + 2))) ^2) + ((Im (a |^ (th1 + 2))) ^2)) is complex real ext-real Element of REAL
|.(a |^ (th1 + 2)).| / ((th1 + 3) !) is complex real ext-real Element of REAL
|.(a |^ (th1 + 2)).| * (((th1 + 3) !) ") is complex real ext-real set
|.(a |^ (th1 + 2)).| / ((th1 + 2) !) is complex real ext-real Element of REAL
|.(a |^ (th1 + 2)).| * (((th1 + 2) !) ") is complex real ext-real set
((Partial_Sums |.(a).|) . th1) + (|.(a |^ (th1 + 2)).| / ((th1 + 3) !)) is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . th1) + (|.(a |^ (th1 + 2)).| / ((th1 + 2) !)) is complex real ext-real Element of REAL
((Partial_Sums |.(a).|) . (th1 + 1)) + (|.(a |^ (th1 + 2)).| / ((th1 + 2) !)) is complex real ext-real Element of REAL
b is complex real ext-real set
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums |.(a).|) . th1 is complex real ext-real Element of REAL
th1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums |.(a).|) . (th1 + 1) is complex real ext-real Element of REAL
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums |.(a).|) . th1 is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
a is complex set
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
a * (Sum (a)) is complex set
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
(Sum (a)) - 1r is complex Element of COMPLEX
- 1r is complex set
(Sum (a)) + (- 1r) is complex set
((Sum (a)) - 1r) - a is complex set
- a is complex set
((Sum (a)) - 1r) + (- a) is complex set
b is complex set
(b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
b (#) (b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
(b) ^\ 2 is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable subsequence of (b)
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(b (#) (b)) . th1 is complex Element of COMPLEX
((b) ^\ 2) . th1 is complex Element of COMPLEX
(b) . th1 is complex Element of COMPLEX
b * ((b) . th1) is complex set
th1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b |^ (th1 + 1) is complex set
b GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(b GeoSeq) . (th1 + 1) is complex Element of COMPLEX
th1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(th1 + 2) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (th1 + 2) is complex real ext-real Element of REAL
(b |^ (th1 + 1)) / ((th1 + 2) !) is complex Element of COMPLEX
((th1 + 2) !) " is non empty complex real ext-real positive non negative set
(b |^ (th1 + 1)) * (((th1 + 2) !) ") is complex set
b * ((b |^ (th1 + 1)) / ((th1 + 2) !)) is complex set
b * (b |^ (th1 + 1)) is complex set
(b * (b |^ (th1 + 1))) / ((th1 + 2) !) is complex Element of COMPLEX
(b * (b |^ (th1 + 1))) * (((th1 + 2) !) ") is complex set
(th1 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b |^ ((th1 + 1) + 1) is complex set
(b GeoSeq) . ((th1 + 1) + 1) is complex Element of COMPLEX
(b |^ ((th1 + 1) + 1)) / ((th1 + 2) !) is complex Element of COMPLEX
(b |^ ((th1 + 1) + 1)) * (((th1 + 2) !) ") is complex set
(b) . (th1 + 2) is complex Element of COMPLEX
(Partial_Sums (a)) . 1 is complex Element of COMPLEX
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a) ^\ (1 + 1) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable subsequence of (a)
Sum ((a) ^\ (1 + 1)) is complex Element of COMPLEX
Partial_Sums ((a) ^\ (1 + 1)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((a) ^\ (1 + 1))) is complex Element of COMPLEX
((Partial_Sums (a)) . 1) + (Sum ((a) ^\ (1 + 1))) is complex Element of COMPLEX
(a) ^\ 2 is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable subsequence of (a)
Sum ((a) ^\ 2) is complex Element of COMPLEX
Partial_Sums ((a) ^\ 2) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((a) ^\ 2)) is complex Element of COMPLEX
((Partial_Sums (a)) . 1) + (Sum ((a) ^\ 2)) is complex Element of COMPLEX
a (#) (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (a (#) (a)) is complex Element of COMPLEX
Partial_Sums (a (#) (a)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a (#) (a))) is complex Element of COMPLEX
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (a)) . (0 + 1) is complex Element of COMPLEX
(Sum (a)) - ((Partial_Sums (a)) . (0 + 1)) is complex Element of COMPLEX
- ((Partial_Sums (a)) . (0 + 1)) is complex set
(Sum (a)) + (- ((Partial_Sums (a)) . (0 + 1))) is complex set
(Partial_Sums (a)) . 0 is complex Element of COMPLEX
(a) . 1 is complex Element of COMPLEX
((Partial_Sums (a)) . 0) + ((a) . 1) is complex Element of COMPLEX
(Sum (a)) - (((Partial_Sums (a)) . 0) + ((a) . 1)) is complex Element of COMPLEX
- (((Partial_Sums (a)) . 0) + ((a) . 1)) is complex set
(Sum (a)) + (- (((Partial_Sums (a)) . 0) + ((a) . 1))) is complex set
(a) . 0 is complex Element of COMPLEX
((a) . 0) + ((a) . 1) is complex Element of COMPLEX
(Sum (a)) - (((a) . 0) + ((a) . 1)) is complex Element of COMPLEX
- (((a) . 0) + ((a) . 1)) is complex set
(Sum (a)) + (- (((a) . 0) + ((a) . 1))) is complex set
1r + ((a) . 1) is complex Element of COMPLEX
(Sum (a)) - (1r + ((a) . 1)) is complex Element of COMPLEX
- (1r + ((a) . 1)) is complex set
(Sum (a)) + (- (1r + ((a) . 1))) is complex set
1r + a is complex set
(Sum (a)) - (1r + a) is complex set
- (1r + a) is complex set
(Sum (a)) + (- (1r + a)) is complex set
b is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
a is complex Element of COMPLEX
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
|.(Sum (a)).| is complex real ext-real Element of REAL
Re (Sum (a)) is complex real ext-real Element of REAL
(Re (Sum (a))) ^2 is complex real ext-real Element of REAL
(Re (Sum (a))) * (Re (Sum (a))) is complex real ext-real set
Im (Sum (a)) is complex real ext-real Element of REAL
(Im (Sum (a))) ^2 is complex real ext-real Element of REAL
(Im (Sum (a))) * (Im (Sum (a))) is complex real ext-real set
((Re (Sum (a))) ^2) + ((Im (Sum (a))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Sum (a))) ^2) + ((Im (Sum (a))) ^2)) is complex real ext-real Element of REAL
|.(a).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum |.(a).| is complex real ext-real Element of REAL
|.(Partial_Sums (a)).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums |.(a).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued non-decreasing Element of K19(K20(NAT,REAL))
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
|.(Partial_Sums (a)).| . b is complex real ext-real Element of REAL
(Partial_Sums |.(a).|) . b is complex real ext-real Element of REAL
(Partial_Sums (a)) . b is complex Element of COMPLEX
|.((Partial_Sums (a)) . b).| is complex real ext-real Element of REAL
Re ((Partial_Sums (a)) . b) is complex real ext-real Element of REAL
(Re ((Partial_Sums (a)) . b)) ^2 is complex real ext-real Element of REAL
(Re ((Partial_Sums (a)) . b)) * (Re ((Partial_Sums (a)) . b)) is complex real ext-real set
Im ((Partial_Sums (a)) . b) is complex real ext-real Element of REAL
(Im ((Partial_Sums (a)) . b)) ^2 is complex real ext-real Element of REAL
(Im ((Partial_Sums (a)) . b)) * (Im ((Partial_Sums (a)) . b)) is complex real ext-real set
((Re ((Partial_Sums (a)) . b)) ^2) + ((Im ((Partial_Sums (a)) . b)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Partial_Sums (a)) . b)) ^2) + ((Im ((Partial_Sums (a)) . b)) ^2)) is complex real ext-real Element of REAL
lim |.(Partial_Sums (a)).| is complex real ext-real Element of REAL
|.(lim (Partial_Sums (a))).| is complex real ext-real Element of REAL
Re (lim (Partial_Sums (a))) is complex real ext-real Element of REAL
(Re (lim (Partial_Sums (a)))) ^2 is complex real ext-real Element of REAL
(Re (lim (Partial_Sums (a)))) * (Re (lim (Partial_Sums (a)))) is complex real ext-real set
Im (lim (Partial_Sums (a))) is complex real ext-real Element of REAL
(Im (lim (Partial_Sums (a)))) ^2 is complex real ext-real Element of REAL
(Im (lim (Partial_Sums (a)))) * (Im (lim (Partial_Sums (a)))) is complex real ext-real set
((Re (lim (Partial_Sums (a)))) ^2) + ((Im (lim (Partial_Sums (a)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (lim (Partial_Sums (a)))) ^2) + ((Im (lim (Partial_Sums (a)))) ^2)) is complex real ext-real Element of REAL
lim (Partial_Sums |.(a).|) is complex real ext-real Element of REAL
a is complex Element of COMPLEX
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
|.(a).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
|.a.| is complex real ext-real Element of REAL
Re a is complex real ext-real Element of REAL
(Re a) ^2 is complex real ext-real Element of REAL
(Re a) * (Re a) is complex real ext-real set
Im a is complex real ext-real Element of REAL
(Im a) ^2 is complex real ext-real Element of REAL
(Im a) * (Im a) is complex real ext-real set
((Re a) ^2) + ((Im a) ^2) is complex real ext-real Element of REAL
sqrt (((Re a) ^2) + ((Im a) ^2)) is complex real ext-real Element of REAL
|.a.| GeoSeq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
|.(a).| . b is complex real ext-real Element of REAL
(|.a.| GeoSeq) . b is complex real ext-real Element of REAL
|.a.| * ((|.a.| GeoSeq) . b) is complex real ext-real Element of REAL
(a) . b is complex Element of COMPLEX
|.((a) . b).| is complex real ext-real Element of REAL
Re ((a) . b) is complex real ext-real Element of REAL
(Re ((a) . b)) ^2 is complex real ext-real Element of REAL
(Re ((a) . b)) * (Re ((a) . b)) is complex real ext-real set
Im ((a) . b) is complex real ext-real Element of REAL
(Im ((a) . b)) ^2 is complex real ext-real Element of REAL
(Im ((a) . b)) * (Im ((a) . b)) is complex real ext-real set
((Re ((a) . b)) ^2) + ((Im ((a) . b)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a) . b)) ^2) + ((Im ((a) . b)) ^2)) is complex real ext-real Element of REAL
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a |^ (b + 1) is complex set
a GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a GeoSeq) . (b + 1) is complex Element of COMPLEX
b + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(b + 2) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (b + 2) is complex real ext-real Element of REAL
(a |^ (b + 1)) / ((b + 2) !) is complex Element of COMPLEX
((b + 2) !) " is non empty complex real ext-real positive non negative set
(a |^ (b + 1)) * (((b + 2) !) ") is complex set
|.((a |^ (b + 1)) / ((b + 2) !)).| is complex real ext-real Element of REAL
Re ((a |^ (b + 1)) / ((b + 2) !)) is complex real ext-real Element of REAL
(Re ((a |^ (b + 1)) / ((b + 2) !))) ^2 is complex real ext-real Element of REAL
(Re ((a |^ (b + 1)) / ((b + 2) !))) * (Re ((a |^ (b + 1)) / ((b + 2) !))) is complex real ext-real set
Im ((a |^ (b + 1)) / ((b + 2) !)) is complex real ext-real Element of REAL
(Im ((a |^ (b + 1)) / ((b + 2) !))) ^2 is complex real ext-real Element of REAL
(Im ((a |^ (b + 1)) / ((b + 2) !))) * (Im ((a |^ (b + 1)) / ((b + 2) !))) is complex real ext-real set
((Re ((a |^ (b + 1)) / ((b + 2) !))) ^2) + ((Im ((a |^ (b + 1)) / ((b + 2) !))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((a |^ (b + 1)) / ((b + 2) !))) ^2) + ((Im ((a |^ (b + 1)) / ((b + 2) !))) ^2)) is complex real ext-real Element of REAL
a #N (b + 1) is complex Element of COMPLEX
|.(a #N (b + 1)).| is complex real ext-real Element of REAL
Re (a #N (b + 1)) is complex real ext-real Element of REAL
(Re (a #N (b + 1))) ^2 is complex real ext-real Element of REAL
(Re (a #N (b + 1))) * (Re (a #N (b + 1))) is complex real ext-real set
Im (a #N (b + 1)) is complex real ext-real Element of REAL
(Im (a #N (b + 1))) ^2 is complex real ext-real Element of REAL
(Im (a #N (b + 1))) * (Im (a #N (b + 1))) is complex real ext-real set
((Re (a #N (b + 1))) ^2) + ((Im (a #N (b + 1))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (a #N (b + 1))) ^2) + ((Im (a #N (b + 1))) ^2)) is complex real ext-real Element of REAL
|.(a #N (b + 1)).| / ((b + 2) !) is complex real ext-real Element of REAL
|.(a #N (b + 1)).| * (((b + 2) !) ") is complex real ext-real set
|.a.| |^ (b + 1) is complex real ext-real Element of REAL
|.a.| GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(|.a.| GeoSeq) . (b + 1) is complex Element of COMPLEX
(|.a.| |^ (b + 1)) / ((b + 2) !) is complex real ext-real Element of REAL
(|.a.| |^ (b + 1)) * (((b + 2) !) ") is complex real ext-real set
|.a.| |^ b is complex real ext-real Element of REAL
(|.a.| GeoSeq) . b is complex Element of COMPLEX
|.a.| * (|.a.| |^ b) is complex real ext-real Element of REAL
(|.a.| |^ (b + 1)) / 1 is complex real ext-real Element of REAL
1 " is non empty complex real ext-real positive non negative set
(|.a.| |^ (b + 1)) * (1 ") is complex real ext-real set
a is complex real ext-real set
b is complex real ext-real Element of REAL
b + 1 is complex real ext-real Element of REAL
b / (b + 1) is complex real ext-real Element of REAL
(b + 1) " is complex real ext-real set
b * ((b + 1) ") is complex real ext-real set
th1 is complex real ext-real Element of REAL
th2 is complex Element of COMPLEX
|.th2.| is complex real ext-real Element of REAL
Re th2 is complex real ext-real Element of REAL
(Re th2) ^2 is complex real ext-real Element of REAL
(Re th2) * (Re th2) is complex real ext-real set
Im th2 is complex real ext-real Element of REAL
(Im th2) ^2 is complex real ext-real Element of REAL
(Im th2) * (Im th2) is complex real ext-real set
((Re th2) ^2) + ((Im th2) ^2) is complex real ext-real Element of REAL
sqrt (((Re th2) ^2) + ((Im th2) ^2)) is complex real ext-real Element of REAL
(th2) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (th2) is complex Element of COMPLEX
Partial_Sums (th2) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (th2)) is complex Element of COMPLEX
|.(Sum (th2)).| is complex real ext-real Element of REAL
Re (Sum (th2)) is complex real ext-real Element of REAL
(Re (Sum (th2))) ^2 is complex real ext-real Element of REAL
(Re (Sum (th2))) * (Re (Sum (th2))) is complex real ext-real set
Im (Sum (th2)) is complex real ext-real Element of REAL
(Im (Sum (th2))) ^2 is complex real ext-real Element of REAL
(Im (Sum (th2))) * (Im (Sum (th2))) is complex real ext-real set
((Re (Sum (th2))) ^2) + ((Im (Sum (th2))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Sum (th2))) ^2) + ((Im (Sum (th2))) ^2)) is complex real ext-real Element of REAL
abs |.th2.| is complex real ext-real Element of REAL
Re |.th2.| is complex real ext-real Element of REAL
(Re |.th2.|) ^2 is complex real ext-real Element of REAL
(Re |.th2.|) * (Re |.th2.|) is complex real ext-real set
Im |.th2.| is complex real ext-real Element of REAL
(Im |.th2.|) ^2 is complex real ext-real Element of REAL
(Im |.th2.|) * (Im |.th2.|) is complex real ext-real set
((Re |.th2.|) ^2) + ((Im |.th2.|) ^2) is complex real ext-real Element of REAL
sqrt (((Re |.th2.|) ^2) + ((Im |.th2.|) ^2)) is complex real ext-real Element of REAL
|.th2.| GeoSeq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (|.th2.| GeoSeq) is complex real ext-real Element of REAL
1 - |.th2.| is complex real ext-real Element of REAL
- |.th2.| is complex real ext-real set
1 + (- |.th2.|) is complex real ext-real set
1 / (1 - |.th2.|) is complex real ext-real Element of REAL
(1 - |.th2.|) " is complex real ext-real set
1 * ((1 - |.th2.|) ") is complex real ext-real set
|.th2.| (#) (|.th2.| GeoSeq) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
|.(th2).| is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
PL is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
|.(th2).| . PL is complex real ext-real Element of REAL
(|.th2.| (#) (|.th2.| GeoSeq)) . PL is complex real ext-real Element of REAL
(|.th2.| GeoSeq) . PL is complex real ext-real Element of REAL
|.th2.| * ((|.th2.| GeoSeq) . PL) is complex real ext-real Element of REAL
PL is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
|.(th2).| . PL is complex real ext-real Element of REAL
(th2) . PL is complex Element of COMPLEX
|.((th2) . PL).| is complex real ext-real Element of REAL
Re ((th2) . PL) is complex real ext-real Element of REAL
(Re ((th2) . PL)) ^2 is complex real ext-real Element of REAL
(Re ((th2) . PL)) * (Re ((th2) . PL)) is complex real ext-real set
Im ((th2) . PL) is complex real ext-real Element of REAL
(Im ((th2) . PL)) ^2 is complex real ext-real Element of REAL
(Im ((th2) . PL)) * (Im ((th2) . PL)) is complex real ext-real set
((Re ((th2) . PL)) ^2) + ((Im ((th2) . PL)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((th2) . PL)) ^2) + ((Im ((th2) . PL)) ^2)) is complex real ext-real Element of REAL
Sum |.(th2).| is complex real ext-real Element of REAL
Sum (|.th2.| (#) (|.th2.| GeoSeq)) is complex real ext-real Element of REAL
|.th2.| / (1 - |.th2.|) is complex real ext-real Element of REAL
|.th2.| * ((1 - |.th2.|) ") is complex real ext-real set
(b / (b + 1)) * (b + 1) is complex real ext-real Element of REAL
|.th2.| * (b + 1) is complex real ext-real Element of REAL
b * |.th2.| is complex real ext-real Element of REAL
b - (b * |.th2.|) is complex real ext-real Element of REAL
- (b * |.th2.|) is complex real ext-real set
b + (- (b * |.th2.|)) is complex real ext-real set
(b * |.th2.|) + |.th2.| is complex real ext-real Element of REAL
((b * |.th2.|) + |.th2.|) - (b * |.th2.|) is complex real ext-real Element of REAL
((b * |.th2.|) + |.th2.|) + (- (b * |.th2.|)) is complex real ext-real set
b * (1 - |.th2.|) is complex real ext-real Element of REAL
(b * (1 - |.th2.|)) / (1 - |.th2.|) is complex real ext-real Element of REAL
(b * (1 - |.th2.|)) * ((1 - |.th2.|) ") is complex real ext-real set
th2 is complex set
|.th2.| is complex real ext-real Element of REAL
Re th2 is complex real ext-real Element of REAL
(Re th2) ^2 is complex real ext-real Element of REAL
(Re th2) * (Re th2) is complex real ext-real set
Im th2 is complex real ext-real Element of REAL
(Im th2) ^2 is complex real ext-real Element of REAL
(Im th2) * (Im th2) is complex real ext-real set
((Re th2) ^2) + ((Im th2) ^2) is complex real ext-real Element of REAL
sqrt (((Re th2) ^2) + ((Im th2) ^2)) is complex real ext-real Element of REAL
(th2) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (th2) is complex Element of COMPLEX
Partial_Sums (th2) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (th2)) is complex Element of COMPLEX
|.(Sum (th2)).| is complex real ext-real Element of REAL
Re (Sum (th2)) is complex real ext-real Element of REAL
(Re (Sum (th2))) ^2 is complex real ext-real Element of REAL
(Re (Sum (th2))) * (Re (Sum (th2))) is complex real ext-real set
Im (Sum (th2)) is complex real ext-real Element of REAL
(Im (Sum (th2))) ^2 is complex real ext-real Element of REAL
(Im (Sum (th2))) * (Im (Sum (th2))) is complex real ext-real set
((Re (Sum (th2))) ^2) + ((Im (Sum (th2))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Sum (th2))) ^2) + ((Im (Sum (th2))) ^2)) is complex real ext-real Element of REAL
b is complex set
a is complex set
b + a is complex set
((b + a)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((b + a)) is complex Element of COMPLEX
Partial_Sums ((b + a)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((b + a))) is complex Element of COMPLEX
(b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (b) is complex Element of COMPLEX
Partial_Sums (b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (b)) is complex Element of COMPLEX
(Sum ((b + a))) - (Sum (b)) is complex Element of COMPLEX
- (Sum (b)) is complex set
(Sum ((b + a))) + (- (Sum (b))) is complex set
(Sum (b)) * a is complex set
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
a * (Sum (a)) is complex set
(a * (Sum (a))) * (Sum (b)) is complex set
((Sum (b)) * a) + ((a * (Sum (a))) * (Sum (b))) is complex set
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
(Sum (b)) * (Sum (a)) is complex Element of COMPLEX
(Sum (b)) * 1r is complex Element of COMPLEX
((Sum (b)) * (Sum (a))) - ((Sum (b)) * 1r) is complex Element of COMPLEX
- ((Sum (b)) * 1r) is complex set
((Sum (b)) * (Sum (a))) + (- ((Sum (b)) * 1r)) is complex set
(Sum (a)) - 1r is complex Element of COMPLEX
- 1r is complex set
(Sum (a)) + (- 1r) is complex set
((Sum (a)) - 1r) - a is complex set
- a is complex set
((Sum (a)) - 1r) + (- a) is complex set
(((Sum (a)) - 1r) - a) + a is complex set
(Sum (b)) * ((((Sum (a)) - 1r) - a) + a) is complex set
(a * (Sum (a))) + a is complex set
(Sum (b)) * ((a * (Sum (a))) + a) is complex set
a is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
a + b is complex real ext-real Element of REAL
() . (a + b) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
(() . (a + b)) - (() . a) is complex real ext-real Element of REAL
- (() . a) is complex real ext-real set
(() . (a + b)) + (- (() . a)) is complex real ext-real set
() . a is complex real ext-real Element of REAL
b * (() . a) is complex real ext-real Element of REAL
- (b * (() . a)) is complex real ext-real Element of REAL
b * <i> is complex set
((b * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum ((b * <i>)) is complex Element of COMPLEX
Partial_Sums ((b * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((b * <i>))) is complex Element of COMPLEX
(() . a) * <i> is complex set
(() . a) + ((() . a) * <i>) is complex set
(Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>)) is complex set
Im ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>))) is complex real ext-real Element of REAL
b * (Im ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>)))) is complex real ext-real Element of REAL
(- (b * (() . a))) - (b * (Im ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>))))) is complex real ext-real Element of REAL
- (b * (Im ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>))))) is complex real ext-real set
(- (b * (() . a))) + (- (b * (Im ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>)))))) is complex real ext-real set
a * <i> is complex set
((a * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((a * <i>)) is complex Element of COMPLEX
Partial_Sums ((a * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((a * <i>))) is complex Element of COMPLEX
Re (Sum ((a * <i>))) is complex real ext-real Element of REAL
(() . (a + b)) - (Re (Sum ((a * <i>)))) is complex real ext-real Element of REAL
- (Re (Sum ((a * <i>)))) is complex real ext-real set
(() . (a + b)) + (- (Re (Sum ((a * <i>))))) is complex real ext-real set
(a + b) * <i> is complex set
(((a + b) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (((a + b) * <i>)) is complex Element of COMPLEX
Partial_Sums (((a + b) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((a + b) * <i>))) is complex Element of COMPLEX
Re (Sum (((a + b) * <i>))) is complex real ext-real Element of REAL
(Re (Sum (((a + b) * <i>)))) - (Re (Sum ((a * <i>)))) is complex real ext-real Element of REAL
(Re (Sum (((a + b) * <i>)))) + (- (Re (Sum ((a * <i>))))) is complex real ext-real set
(a * <i>) + (b * <i>) is complex set
(((a * <i>) + (b * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (((a * <i>) + (b * <i>))) is complex Element of COMPLEX
Partial_Sums (((a * <i>) + (b * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((a * <i>) + (b * <i>)))) is complex Element of COMPLEX
(Sum (((a * <i>) + (b * <i>)))) - (Sum ((a * <i>))) is complex Element of COMPLEX
- (Sum ((a * <i>))) is complex set
(Sum (((a * <i>) + (b * <i>)))) + (- (Sum ((a * <i>)))) is complex set
Re ((Sum (((a * <i>) + (b * <i>)))) - (Sum ((a * <i>)))) is complex real ext-real Element of REAL
(Sum ((a * <i>))) * (b * <i>) is complex set
(b * <i>) * (Sum ((b * <i>))) is complex set
((b * <i>) * (Sum ((b * <i>)))) * (Sum ((a * <i>))) is complex set
((Sum ((a * <i>))) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * (Sum ((a * <i>)))) is complex set
Re (((Sum ((a * <i>))) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * (Sum ((a * <i>))))) is complex real ext-real Element of REAL
((() . a) + ((() . a) * <i>)) * (b * <i>) is complex set
(((() . a) + ((() . a) * <i>)) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * (Sum ((a * <i>)))) is complex set
Re ((((() . a) + ((() . a) * <i>)) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * (Sum ((a * <i>))))) is complex real ext-real Element of REAL
((b * <i>) * (Sum ((b * <i>)))) * ((() . a) + ((() . a) * <i>)) is complex set
(((() . a) + ((() . a) * <i>)) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * ((() . a) + ((() . a) * <i>))) is complex set
Re ((((() . a) + ((() . a) * <i>)) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * ((() . a) + ((() . a) * <i>)))) is complex real ext-real Element of REAL
b * (() . a) is complex real ext-real Element of REAL
(b * (() . a)) * <i> is complex set
(- (b * (() . a))) + ((b * (() . a)) * <i>) is complex set
Re ((- (b * (() . a))) + ((b * (() . a)) * <i>)) is complex real ext-real Element of REAL
Re (((b * <i>) * (Sum ((b * <i>)))) * ((() . a) + ((() . a) * <i>))) is complex real ext-real Element of REAL
(Re ((- (b * (() . a))) + ((b * (() . a)) * <i>))) + (Re (((b * <i>) * (Sum ((b * <i>)))) * ((() . a) + ((() . a) * <i>)))) is complex real ext-real Element of REAL
(b * <i>) * ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>))) is complex set
Re ((b * <i>) * ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>)))) is complex real ext-real Element of REAL
(- (b * (() . a))) + (Re ((b * <i>) * ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>))))) is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
a + b is complex real ext-real Element of REAL
() . (a + b) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
(() . (a + b)) - (() . a) is complex real ext-real Element of REAL
- (() . a) is complex real ext-real set
(() . (a + b)) + (- (() . a)) is complex real ext-real set
() . a is complex real ext-real Element of REAL
b * (() . a) is complex real ext-real Element of REAL
b * <i> is complex set
((b * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum ((b * <i>)) is complex Element of COMPLEX
Partial_Sums ((b * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((b * <i>))) is complex Element of COMPLEX
(() . a) * <i> is complex set
(() . a) + ((() . a) * <i>) is complex set
(Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>)) is complex set
Re ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>))) is complex real ext-real Element of REAL
b * (Re ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>)))) is complex real ext-real Element of REAL
(b * (() . a)) + (b * (Re ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>))))) is complex real ext-real Element of REAL
a * <i> is complex set
((a * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((a * <i>)) is complex Element of COMPLEX
Partial_Sums ((a * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((a * <i>))) is complex Element of COMPLEX
Im (Sum ((a * <i>))) is complex real ext-real Element of REAL
(() . (a + b)) - (Im (Sum ((a * <i>)))) is complex real ext-real Element of REAL
- (Im (Sum ((a * <i>)))) is complex real ext-real set
(() . (a + b)) + (- (Im (Sum ((a * <i>))))) is complex real ext-real set
(a + b) * <i> is complex set
(((a + b) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (((a + b) * <i>)) is complex Element of COMPLEX
Partial_Sums (((a + b) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((a + b) * <i>))) is complex Element of COMPLEX
Im (Sum (((a + b) * <i>))) is complex real ext-real Element of REAL
(Im (Sum (((a + b) * <i>)))) - (Im (Sum ((a * <i>)))) is complex real ext-real Element of REAL
(Im (Sum (((a + b) * <i>)))) + (- (Im (Sum ((a * <i>))))) is complex real ext-real set
(a * <i>) + (b * <i>) is complex set
(((a * <i>) + (b * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (((a * <i>) + (b * <i>))) is complex Element of COMPLEX
Partial_Sums (((a * <i>) + (b * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((a * <i>) + (b * <i>)))) is complex Element of COMPLEX
(Sum (((a * <i>) + (b * <i>)))) - (Sum ((a * <i>))) is complex Element of COMPLEX
- (Sum ((a * <i>))) is complex set
(Sum (((a * <i>) + (b * <i>)))) + (- (Sum ((a * <i>)))) is complex set
Im ((Sum (((a * <i>) + (b * <i>)))) - (Sum ((a * <i>)))) is complex real ext-real Element of REAL
(Sum ((a * <i>))) * (b * <i>) is complex set
(b * <i>) * (Sum ((b * <i>))) is complex set
((b * <i>) * (Sum ((b * <i>)))) * (Sum ((a * <i>))) is complex set
((Sum ((a * <i>))) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * (Sum ((a * <i>)))) is complex set
Im (((Sum ((a * <i>))) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * (Sum ((a * <i>))))) is complex real ext-real Element of REAL
((() . a) + ((() . a) * <i>)) * (b * <i>) is complex set
(((() . a) + ((() . a) * <i>)) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * (Sum ((a * <i>)))) is complex set
Im ((((() . a) + ((() . a) * <i>)) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * (Sum ((a * <i>))))) is complex real ext-real Element of REAL
((b * <i>) * (Sum ((b * <i>)))) * ((() . a) + ((() . a) * <i>)) is complex set
(((() . a) + ((() . a) * <i>)) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * ((() . a) + ((() . a) * <i>))) is complex set
Im ((((() . a) + ((() . a) * <i>)) * (b * <i>)) + (((b * <i>) * (Sum ((b * <i>)))) * ((() . a) + ((() . a) * <i>)))) is complex real ext-real Element of REAL
b * (() . a) is complex real ext-real Element of REAL
- (b * (() . a)) is complex real ext-real Element of REAL
(b * (() . a)) * <i> is complex set
(- (b * (() . a))) + ((b * (() . a)) * <i>) is complex set
Im ((- (b * (() . a))) + ((b * (() . a)) * <i>)) is complex real ext-real Element of REAL
Im (((b * <i>) * (Sum ((b * <i>)))) * ((() . a) + ((() . a) * <i>))) is complex real ext-real Element of REAL
(Im ((- (b * (() . a))) + ((b * (() . a)) * <i>))) + (Im (((b * <i>) * (Sum ((b * <i>)))) * ((() . a) + ((() . a) * <i>)))) is complex real ext-real Element of REAL
(b * <i>) * ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>))) is complex set
Im ((b * <i>) * ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>)))) is complex real ext-real Element of REAL
(b * (() . a)) + (Im ((b * <i>) * ((Sum ((b * <i>))) * ((() . a) + ((() . a) * <i>))))) is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
a + b is complex real ext-real Element of REAL
() . (a + b) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
(() . (a + b)) - (() . a) is complex real ext-real Element of REAL
- (() . a) is complex real ext-real set
(() . (a + b)) + (- (() . a)) is complex real ext-real set
b * (() . a) is complex real ext-real Element of REAL
(b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (b) is complex Element of COMPLEX
Partial_Sums (b) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (b)) is complex Element of COMPLEX
Re (Sum (b)) is complex real ext-real Element of REAL
(b * (() . a)) * (Re (Sum (b))) is complex real ext-real Element of REAL
(b * (() . a)) + ((b * (() . a)) * (Re (Sum (b)))) is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (a) is complex Element of COMPLEX
Partial_Sums (a) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (a)) is complex Element of COMPLEX
Re (Sum (a)) is complex real ext-real Element of REAL
(() . (a + b)) - (Re (Sum (a))) is complex real ext-real Element of REAL
- (Re (Sum (a))) is complex real ext-real set
(() . (a + b)) + (- (Re (Sum (a)))) is complex real ext-real set
((a + b)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((a + b)) is complex Element of COMPLEX
Partial_Sums ((a + b)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((a + b))) is complex Element of COMPLEX
Re (Sum ((a + b))) is complex real ext-real Element of REAL
(Re (Sum ((a + b)))) - (Re (Sum (a))) is complex real ext-real Element of REAL
(Re (Sum ((a + b)))) + (- (Re (Sum (a)))) is complex real ext-real set
(Sum ((a + b))) - (Sum (a)) is complex Element of COMPLEX
- (Sum (a)) is complex set
(Sum ((a + b))) + (- (Sum (a))) is complex set
Re ((Sum ((a + b))) - (Sum (a))) is complex real ext-real Element of REAL
(Sum (a)) * b is complex set
b * (Sum (b)) is complex set
(b * (Sum (b))) * (Sum (a)) is complex set
((Sum (a)) * b) + ((b * (Sum (b))) * (Sum (a))) is complex set
Re (((Sum (a)) * b) + ((b * (Sum (b))) * (Sum (a)))) is complex real ext-real Element of REAL
Re ((Sum (a)) * b) is complex real ext-real Element of REAL
Re ((b * (Sum (b))) * (Sum (a))) is complex real ext-real Element of REAL
(Re ((Sum (a)) * b)) + (Re ((b * (Sum (b))) * (Sum (a)))) is complex real ext-real Element of REAL
b * (Re (Sum (a))) is complex real ext-real Element of REAL
(b * (Re (Sum (a)))) + (Re ((b * (Sum (b))) * (Sum (a)))) is complex real ext-real Element of REAL
(Sum (b)) * (Sum (a)) is complex Element of COMPLEX
b * ((Sum (b)) * (Sum (a))) is complex set
Re (b * ((Sum (b)) * (Sum (a)))) is complex real ext-real Element of REAL
(b * (() . a)) + (Re (b * ((Sum (b)) * (Sum (a))))) is complex real ext-real Element of REAL
Re ((Sum (b)) * (Sum (a))) is complex real ext-real Element of REAL
b * (Re ((Sum (b)) * (Sum (a)))) is complex real ext-real Element of REAL
(b * (() . a)) + (b * (Re ((Sum (b)) * (Sum (a))))) is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (a) is complex real ext-real Element of REAL
(Sum (b)) * (Sum (a)) is complex set
Re ((Sum (b)) * (Sum (a))) is complex real ext-real Element of REAL
b * (Re ((Sum (b)) * (Sum (a)))) is complex real ext-real Element of REAL
(b * (() . a)) + (b * (Re ((Sum (b)) * (Sum (a))))) is complex real ext-real Element of REAL
(() . a) + (0 * <i>) is complex set
(Sum (b)) * ((() . a) + (0 * <i>)) is complex set
Re ((Sum (b)) * ((() . a) + (0 * <i>))) is complex real ext-real Element of REAL
b * (Re ((Sum (b)) * ((() . a) + (0 * <i>)))) is complex real ext-real Element of REAL
(b * (() . a)) + (b * (Re ((Sum (b)) * ((() . a) + (0 * <i>))))) is complex real ext-real Element of REAL
(() . a) * (Re (Sum (b))) is complex real ext-real Element of REAL
b * ((() . a) * (Re (Sum (b)))) is complex real ext-real Element of REAL
(b * (() . a)) + (b * ((() . a) * (Re (Sum (b))))) is complex real ext-real Element of REAL
a is complex real ext-real set
diff ((),a) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
- (() . a) is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
() . b is complex real ext-real Element of REAL
() . b is complex real ext-real Element of REAL
(() . b) * <i> is complex set
(() . b) + ((() . b) * <i>) is complex set
th1 is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
th2 is Relation-like non-empty NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent 0 -convergent bounded_above bounded_below Element of K19(K20(NAT,REAL))
th2 " is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
th1 /* th2 is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(th2 ") (#) (th1 /* th2) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((th2 ") (#) (th1 /* th2)) is complex real ext-real Element of REAL
PL is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((th2 ") (#) (th1 /* th2)) . PL is complex real ext-real Element of REAL
th2 . PL is complex real ext-real Element of REAL
(th2 . PL) * <i> is complex set
(((th2 . PL) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((th2 . PL) * <i>)) is complex Element of COMPLEX
Partial_Sums (((th2 . PL) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((th2 . PL) * <i>))) is complex Element of COMPLEX
(Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
Im ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
- (Im ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
(th2 ") . PL is complex real ext-real Element of REAL
(th1 /* th2) . PL is complex real ext-real Element of REAL
((th2 ") . PL) * ((th1 /* th2) . PL) is complex real ext-real Element of REAL
(th2 . PL) " is complex real ext-real Element of REAL
((th2 . PL) ") * ((th1 /* th2) . PL) is complex real ext-real Element of REAL
dom th1 is V46() V47() V48() Element of K19(REAL)
rng th2 is V46() V47() V48() Element of K19(REAL)
th1 . (th2 . PL) is complex real ext-real Element of REAL
((th2 . PL) ") * (th1 . (th2 . PL)) is complex real ext-real Element of REAL
(th2 . PL) * (Im ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
- ((th2 . PL) * (Im ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>))))) is complex real ext-real Element of REAL
((th2 . PL) ") * (- ((th2 . PL) * (Im ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>)))))) is complex real ext-real Element of REAL
((th2 . PL) ") * (th2 . PL) is complex real ext-real Element of REAL
(((th2 . PL) ") * (th2 . PL)) * (Im ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
- ((((th2 . PL) ") * (th2 . PL)) * (Im ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>))))) is complex real ext-real Element of REAL
1 * (Im ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
- (1 * (Im ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>))))) is complex real ext-real Element of REAL
PL is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
c₈ is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
p is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
lim th2 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of REAL
c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₁₂ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₈ . c₁₂ is complex Element of COMPLEX
(c₈ . c₁₂) - 0c is complex Element of COMPLEX
- 0c is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
(c₈ . c₁₂) + (- 0c) is complex set
|.((c₈ . c₁₂) - 0c).| is complex real ext-real Element of REAL
Re ((c₈ . c₁₂) - 0c) is complex real ext-real Element of REAL
(Re ((c₈ . c₁₂) - 0c)) ^2 is complex real ext-real Element of REAL
(Re ((c₈ . c₁₂) - 0c)) * (Re ((c₈ . c₁₂) - 0c)) is complex real ext-real set
Im ((c₈ . c₁₂) - 0c) is complex real ext-real Element of REAL
(Im ((c₈ . c₁₂) - 0c)) ^2 is complex real ext-real Element of REAL
(Im ((c₈ . c₁₂) - 0c)) * (Im ((c₈ . c₁₂) - 0c)) is complex real ext-real set
((Re ((c₈ . c₁₂) - 0c)) ^2) + ((Im ((c₈ . c₁₂) - 0c)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((c₈ . c₁₂) - 0c)) ^2) + ((Im ((c₈ . c₁₂) - 0c)) ^2)) is complex real ext-real Element of REAL
th2 . c₁₂ is complex real ext-real Element of REAL
(th2 . c₁₂) * <i> is complex set
(((th2 . c₁₂) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((th2 . c₁₂) * <i>)) is complex Element of COMPLEX
Partial_Sums (((th2 . c₁₂) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((th2 . c₁₂) * <i>))) is complex Element of COMPLEX
(Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
|.((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>))).| is complex real ext-real Element of REAL
Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
(Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2 is complex real ext-real Element of REAL
(Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) * (Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real set
Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
(Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2 is complex real ext-real Element of REAL
(Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) * (Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real set
((Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2) + ((Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2) + ((Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2)) is complex real ext-real Element of REAL
|.(Sum (((th2 . c₁₂) * <i>))).| is complex real ext-real Element of REAL
Re (Sum (((th2 . c₁₂) * <i>))) is complex real ext-real Element of REAL
(Re (Sum (((th2 . c₁₂) * <i>)))) ^2 is complex real ext-real Element of REAL
(Re (Sum (((th2 . c₁₂) * <i>)))) * (Re (Sum (((th2 . c₁₂) * <i>)))) is complex real ext-real set
Im (Sum (((th2 . c₁₂) * <i>))) is complex real ext-real Element of REAL
(Im (Sum (((th2 . c₁₂) * <i>)))) ^2 is complex real ext-real Element of REAL
(Im (Sum (((th2 . c₁₂) * <i>)))) * (Im (Sum (((th2 . c₁₂) * <i>)))) is complex real ext-real set
((Re (Sum (((th2 . c₁₂) * <i>)))) ^2) + ((Im (Sum (((th2 . c₁₂) * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Sum (((th2 . c₁₂) * <i>)))) ^2) + ((Im (Sum (((th2 . c₁₂) * <i>)))) ^2)) is complex real ext-real Element of REAL
|.((() . b) + ((() . b) * <i>)).| is complex real ext-real Element of REAL
Re ((() . b) + ((() . b) * <i>)) is complex real ext-real Element of REAL
(Re ((() . b) + ((() . b) * <i>))) ^2 is complex real ext-real Element of REAL
(Re ((() . b) + ((() . b) * <i>))) * (Re ((() . b) + ((() . b) * <i>))) is complex real ext-real set
Im ((() . b) + ((() . b) * <i>)) is complex real ext-real Element of REAL
(Im ((() . b) + ((() . b) * <i>))) ^2 is complex real ext-real Element of REAL
(Im ((() . b) + ((() . b) * <i>))) * (Im ((() . b) + ((() . b) * <i>))) is complex real ext-real set
((Re ((() . b) + ((() . b) * <i>))) ^2) + ((Im ((() . b) + ((() . b) * <i>))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((() . b) + ((() . b) * <i>))) ^2) + ((Im ((() . b) + ((() . b) * <i>))) ^2)) is complex real ext-real Element of REAL
|.(Sum (((th2 . c₁₂) * <i>))).| * |.((() . b) + ((() . b) * <i>)).| is complex real ext-real Element of REAL
b * <i> is complex set
((b * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((b * <i>)) is complex Element of COMPLEX
Partial_Sums ((b * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((b * <i>))) is complex Element of COMPLEX
|.(Sum ((b * <i>))).| is complex real ext-real Element of REAL
Re (Sum ((b * <i>))) is complex real ext-real Element of REAL
(Re (Sum ((b * <i>)))) ^2 is complex real ext-real Element of REAL
(Re (Sum ((b * <i>)))) * (Re (Sum ((b * <i>)))) is complex real ext-real set
Im (Sum ((b * <i>))) is complex real ext-real Element of REAL
(Im (Sum ((b * <i>)))) ^2 is complex real ext-real Element of REAL
(Im (Sum ((b * <i>)))) * (Im (Sum ((b * <i>)))) is complex real ext-real set
((Re (Sum ((b * <i>)))) ^2) + ((Im (Sum ((b * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Sum ((b * <i>)))) ^2) + ((Im (Sum ((b * <i>)))) ^2)) is complex real ext-real Element of REAL
|.(Sum (((th2 . c₁₂) * <i>))).| * |.(Sum ((b * <i>))).| is complex real ext-real Element of REAL
|.(Sum (((th2 . c₁₂) * <i>))).| * 1 is complex real ext-real Element of REAL
(th2 . c₁₂) - 0 is complex real ext-real Element of REAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
(th2 . c₁₂) + (- 0) is complex real ext-real set
abs ((th2 . c₁₂) - 0) is complex real ext-real Element of REAL
Re ((th2 . c₁₂) - 0) is complex real ext-real Element of REAL
(Re ((th2 . c₁₂) - 0)) ^2 is complex real ext-real Element of REAL
(Re ((th2 . c₁₂) - 0)) * (Re ((th2 . c₁₂) - 0)) is complex real ext-real set
Im ((th2 . c₁₂) - 0) is complex real ext-real Element of REAL
(Im ((th2 . c₁₂) - 0)) ^2 is complex real ext-real Element of REAL
(Im ((th2 . c₁₂) - 0)) * (Im ((th2 . c₁₂) - 0)) is complex real ext-real set
((Re ((th2 . c₁₂) - 0)) ^2) + ((Im ((th2 . c₁₂) - 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((th2 . c₁₂) - 0)) ^2) + ((Im ((th2 . c₁₂) - 0)) ^2)) is complex real ext-real Element of REAL
0 + ((th2 . c₁₂) * <i>) is complex set
Re ((th2 . c₁₂) * <i>) is complex real ext-real Element of REAL
Im ((th2 . c₁₂) * <i>) is complex real ext-real Element of REAL
|.((th2 . c₁₂) * <i>).| is complex real ext-real Element of REAL
(Re ((th2 . c₁₂) * <i>)) ^2 is complex real ext-real Element of REAL
(Re ((th2 . c₁₂) * <i>)) * (Re ((th2 . c₁₂) * <i>)) is complex real ext-real set
(Im ((th2 . c₁₂) * <i>)) ^2 is complex real ext-real Element of REAL
(Im ((th2 . c₁₂) * <i>)) * (Im ((th2 . c₁₂) * <i>)) is complex real ext-real set
((Re ((th2 . c₁₂) * <i>)) ^2) + ((Im ((th2 . c₁₂) * <i>)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((th2 . c₁₂) * <i>)) ^2) + ((Im ((th2 . c₁₂) * <i>)) ^2)) is complex real ext-real Element of REAL
abs (th2 . c₁₂) is complex real ext-real Element of REAL
Re (th2 . c₁₂) is complex real ext-real Element of REAL
(Re (th2 . c₁₂)) ^2 is complex real ext-real Element of REAL
(Re (th2 . c₁₂)) * (Re (th2 . c₁₂)) is complex real ext-real set
Im (th2 . c₁₂) is complex real ext-real Element of REAL
(Im (th2 . c₁₂)) ^2 is complex real ext-real Element of REAL
(Im (th2 . c₁₂)) * (Im (th2 . c₁₂)) is complex real ext-real set
((Re (th2 . c₁₂)) ^2) + ((Im (th2 . c₁₂)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (th2 . c₁₂)) ^2) + ((Im (th2 . c₁₂)) ^2)) is complex real ext-real Element of REAL
c₁₂ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₈ . c₁₂ is complex Element of COMPLEX
(c₈ . c₁₂) - 0c is complex Element of COMPLEX
(c₈ . c₁₂) + (- 0c) is complex set
|.((c₈ . c₁₂) - 0c).| is complex real ext-real Element of REAL
Re ((c₈ . c₁₂) - 0c) is complex real ext-real Element of REAL
(Re ((c₈ . c₁₂) - 0c)) ^2 is complex real ext-real Element of REAL
(Re ((c₈ . c₁₂) - 0c)) * (Re ((c₈ . c₁₂) - 0c)) is complex real ext-real set
Im ((c₈ . c₁₂) - 0c) is complex real ext-real Element of REAL
(Im ((c₈ . c₁₂) - 0c)) ^2 is complex real ext-real Element of REAL
(Im ((c₈ . c₁₂) - 0c)) * (Im ((c₈ . c₁₂) - 0c)) is complex real ext-real set
((Re ((c₈ . c₁₂) - 0c)) ^2) + ((Im ((c₈ . c₁₂) - 0c)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((c₈ . c₁₂) - 0c)) ^2) + ((Im ((c₈ . c₁₂) - 0c)) ^2)) is complex real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
lim c₈ is complex Element of COMPLEX
- c₈ is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(- 1) (#) c₈ is Relation-like NAT -defined Function-like total complex-valued set
lim (- c₈) is complex Element of COMPLEX
- 0c is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of COMPLEX
Im (- c₈) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
p is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Im (- c₈)) . p is complex real ext-real Element of REAL
th2 . p is complex real ext-real Element of REAL
(th2 . p) * <i> is complex set
(((th2 . p) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((th2 . p) * <i>)) is complex Element of COMPLEX
Partial_Sums (((th2 . p) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((th2 . p) * <i>))) is complex Element of COMPLEX
(Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
Im ((Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
- (Im ((Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
(- c₈) . p is complex Element of COMPLEX
Im ((- c₈) . p) is complex real ext-real Element of REAL
c₈ . p is complex Element of COMPLEX
- (c₈ . p) is complex Element of COMPLEX
Im (- (c₈ . p)) is complex real ext-real Element of REAL
- ((Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex set
Im (- ((Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Im (- c₈)) . p is complex real ext-real Element of REAL
PL . p is complex real ext-real Element of REAL
th2 . p is complex real ext-real Element of REAL
(th2 . p) * <i> is complex set
(((th2 . p) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((th2 . p) * <i>)) is complex Element of COMPLEX
Partial_Sums (((th2 . p) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((th2 . p) * <i>))) is complex Element of COMPLEX
(Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
Im ((Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
- (Im ((Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
PL . p is complex real ext-real Element of REAL
((th2 ") (#) (th1 /* th2)) . p is complex real ext-real Element of REAL
th2 . p is complex real ext-real Element of REAL
(th2 . p) * <i> is complex set
(((th2 . p) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((th2 . p) * <i>)) is complex Element of COMPLEX
Partial_Sums (((th2 . p) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((th2 . p) * <i>))) is complex Element of COMPLEX
(Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
Im ((Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
- (Im ((Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
PL is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
- (() . b) is complex real ext-real Element of REAL
c₈ is complex real ext-real Element of REAL
PL . c₈ is complex real ext-real Element of REAL
(- (() . b)) * c₈ is complex real ext-real Element of REAL
c₈ * (() . b) is complex real ext-real Element of REAL
- (c₈ * (() . b)) is complex real ext-real Element of REAL
c₈ is complex real ext-real Element of REAL
PL . c₈ is complex real ext-real Element of REAL
(- (() . b)) * c₈ is complex real ext-real Element of REAL
p is complex real ext-real Element of REAL
c₈ is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued linear Element of K19(K20(REAL,REAL))
th2 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued RestFunc-like Element of K19(K20(REAL,REAL))
b - 1 is complex real ext-real Element of REAL
b + (- 1) is complex real ext-real set
b + 1 is complex real ext-real Element of REAL
].(b - 1),(b + 1).[ is open V46() V47() V48() Element of K19(REAL)
p is complex real ext-real Element of REAL
() . p is complex real ext-real Element of REAL
(() . p) - (() . b) is complex real ext-real Element of REAL
- (() . b) is complex real ext-real set
(() . p) + (- (() . b)) is complex real ext-real set
p - b is complex real ext-real Element of REAL
- b is complex real ext-real set
p + (- b) is complex real ext-real set
c₈ . (p - b) is complex real ext-real Element of REAL
th2 . (p - b) is complex real ext-real Element of REAL
(c₈ . (p - b)) + (th2 . (p - b)) is complex real ext-real Element of REAL
b + (p - b) is complex real ext-real Element of REAL
(p - b) * (() . b) is complex real ext-real Element of REAL
- ((p - b) * (() . b)) is complex real ext-real Element of REAL
(p - b) * <i> is complex set
(((p - b) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((p - b) * <i>)) is complex Element of COMPLEX
Partial_Sums (((p - b) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((p - b) * <i>))) is complex Element of COMPLEX
(Sum (((p - b) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
Im ((Sum (((p - b) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
(p - b) * (Im ((Sum (((p - b) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
(- ((p - b) * (() . b))) - ((p - b) * (Im ((Sum (((p - b) * <i>))) * ((() . b) + ((() . b) * <i>))))) is complex real ext-real Element of REAL
- ((p - b) * (Im ((Sum (((p - b) * <i>))) * ((() . b) + ((() . b) * <i>))))) is complex real ext-real set
(- ((p - b) * (() . b))) + (- ((p - b) * (Im ((Sum (((p - b) * <i>))) * ((() . b) + ((() . b) * <i>)))))) is complex real ext-real set
th1 . (p - b) is complex real ext-real Element of REAL
(- ((p - b) * (() . b))) + (th1 . (p - b)) is complex real ext-real Element of REAL
p is complex real ext-real Element of REAL
() . p is complex real ext-real Element of REAL
(() . p) - (() . b) is complex real ext-real Element of REAL
- (() . b) is complex real ext-real set
(() . p) + (- (() . b)) is complex real ext-real set
p - b is complex real ext-real Element of REAL
- b is complex real ext-real set
p + (- b) is complex real ext-real set
c₈ . (p - b) is complex real ext-real Element of REAL
th2 . (p - b) is complex real ext-real Element of REAL
(c₈ . (p - b)) + (th2 . (p - b)) is complex real ext-real Element of REAL
p is open V46() V47() V48() Neighbourhood of b
c₈ . 1 is complex real ext-real Element of REAL
1 * (() . b) is complex real ext-real Element of REAL
- (1 * (() . b)) is complex real ext-real Element of REAL
- (() . b) is complex real ext-real Element of REAL
p is open V46() V47() V48() Neighbourhood of b
a is complex real ext-real set
diff ((),a) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
() . b is complex real ext-real Element of REAL
() . b is complex real ext-real Element of REAL
(() . b) * <i> is complex set
(() . b) + ((() . b) * <i>) is complex set
th1 is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
th2 is Relation-like non-empty NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent 0 -convergent bounded_above bounded_below Element of K19(K20(NAT,REAL))
th2 " is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
th1 /* th2 is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(th2 ") (#) (th1 /* th2) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((th2 ") (#) (th1 /* th2)) is complex real ext-real Element of REAL
PL is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((th2 ") (#) (th1 /* th2)) . PL is complex real ext-real Element of REAL
th2 . PL is complex real ext-real Element of REAL
(th2 . PL) * <i> is complex set
(((th2 . PL) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((th2 . PL) * <i>)) is complex Element of COMPLEX
Partial_Sums (((th2 . PL) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((th2 . PL) * <i>))) is complex Element of COMPLEX
(Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
Re ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
(th2 ") . PL is complex real ext-real Element of REAL
(th1 /* th2) . PL is complex real ext-real Element of REAL
((th2 ") . PL) * ((th1 /* th2) . PL) is complex real ext-real Element of REAL
(th2 . PL) " is complex real ext-real Element of REAL
((th2 . PL) ") * ((th1 /* th2) . PL) is complex real ext-real Element of REAL
dom th1 is V46() V47() V48() Element of K19(REAL)
rng th2 is V46() V47() V48() Element of K19(REAL)
th1 . (th2 . PL) is complex real ext-real Element of REAL
((th2 . PL) ") * (th1 . (th2 . PL)) is complex real ext-real Element of REAL
(th2 . PL) * (Re ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
((th2 . PL) ") * ((th2 . PL) * (Re ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>))))) is complex real ext-real Element of REAL
((th2 . PL) ") * (th2 . PL) is complex real ext-real Element of REAL
(((th2 . PL) ") * (th2 . PL)) * (Re ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
1 * (Re ((Sum (((th2 . PL) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
PL is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
c₈ is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
p is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
lim th2 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of REAL
c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₁₂ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₈ . c₁₂ is complex Element of COMPLEX
(c₈ . c₁₂) - 0c is complex Element of COMPLEX
- 0c is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
(c₈ . c₁₂) + (- 0c) is complex set
|.((c₈ . c₁₂) - 0c).| is complex real ext-real Element of REAL
Re ((c₈ . c₁₂) - 0c) is complex real ext-real Element of REAL
(Re ((c₈ . c₁₂) - 0c)) ^2 is complex real ext-real Element of REAL
(Re ((c₈ . c₁₂) - 0c)) * (Re ((c₈ . c₁₂) - 0c)) is complex real ext-real set
Im ((c₈ . c₁₂) - 0c) is complex real ext-real Element of REAL
(Im ((c₈ . c₁₂) - 0c)) ^2 is complex real ext-real Element of REAL
(Im ((c₈ . c₁₂) - 0c)) * (Im ((c₈ . c₁₂) - 0c)) is complex real ext-real set
((Re ((c₈ . c₁₂) - 0c)) ^2) + ((Im ((c₈ . c₁₂) - 0c)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((c₈ . c₁₂) - 0c)) ^2) + ((Im ((c₈ . c₁₂) - 0c)) ^2)) is complex real ext-real Element of REAL
th2 . c₁₂ is complex real ext-real Element of REAL
(th2 . c₁₂) * <i> is complex set
(((th2 . c₁₂) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((th2 . c₁₂) * <i>)) is complex Element of COMPLEX
Partial_Sums (((th2 . c₁₂) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((th2 . c₁₂) * <i>))) is complex Element of COMPLEX
(Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
|.((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>))).| is complex real ext-real Element of REAL
Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
(Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2 is complex real ext-real Element of REAL
(Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) * (Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real set
Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
(Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2 is complex real ext-real Element of REAL
(Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) * (Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real set
((Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2) + ((Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2) + ((Im ((Sum (((th2 . c₁₂) * <i>))) * ((() . b) + ((() . b) * <i>)))) ^2)) is complex real ext-real Element of REAL
|.(Sum (((th2 . c₁₂) * <i>))).| is complex real ext-real Element of REAL
Re (Sum (((th2 . c₁₂) * <i>))) is complex real ext-real Element of REAL
(Re (Sum (((th2 . c₁₂) * <i>)))) ^2 is complex real ext-real Element of REAL
(Re (Sum (((th2 . c₁₂) * <i>)))) * (Re (Sum (((th2 . c₁₂) * <i>)))) is complex real ext-real set
Im (Sum (((th2 . c₁₂) * <i>))) is complex real ext-real Element of REAL
(Im (Sum (((th2 . c₁₂) * <i>)))) ^2 is complex real ext-real Element of REAL
(Im (Sum (((th2 . c₁₂) * <i>)))) * (Im (Sum (((th2 . c₁₂) * <i>)))) is complex real ext-real set
((Re (Sum (((th2 . c₁₂) * <i>)))) ^2) + ((Im (Sum (((th2 . c₁₂) * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Sum (((th2 . c₁₂) * <i>)))) ^2) + ((Im (Sum (((th2 . c₁₂) * <i>)))) ^2)) is complex real ext-real Element of REAL
|.((() . b) + ((() . b) * <i>)).| is complex real ext-real Element of REAL
Re ((() . b) + ((() . b) * <i>)) is complex real ext-real Element of REAL
(Re ((() . b) + ((() . b) * <i>))) ^2 is complex real ext-real Element of REAL
(Re ((() . b) + ((() . b) * <i>))) * (Re ((() . b) + ((() . b) * <i>))) is complex real ext-real set
Im ((() . b) + ((() . b) * <i>)) is complex real ext-real Element of REAL
(Im ((() . b) + ((() . b) * <i>))) ^2 is complex real ext-real Element of REAL
(Im ((() . b) + ((() . b) * <i>))) * (Im ((() . b) + ((() . b) * <i>))) is complex real ext-real set
((Re ((() . b) + ((() . b) * <i>))) ^2) + ((Im ((() . b) + ((() . b) * <i>))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((() . b) + ((() . b) * <i>))) ^2) + ((Im ((() . b) + ((() . b) * <i>))) ^2)) is complex real ext-real Element of REAL
|.(Sum (((th2 . c₁₂) * <i>))).| * |.((() . b) + ((() . b) * <i>)).| is complex real ext-real Element of REAL
b * <i> is complex set
((b * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((b * <i>)) is complex Element of COMPLEX
Partial_Sums ((b * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((b * <i>))) is complex Element of COMPLEX
|.(Sum ((b * <i>))).| is complex real ext-real Element of REAL
Re (Sum ((b * <i>))) is complex real ext-real Element of REAL
(Re (Sum ((b * <i>)))) ^2 is complex real ext-real Element of REAL
(Re (Sum ((b * <i>)))) * (Re (Sum ((b * <i>)))) is complex real ext-real set
Im (Sum ((b * <i>))) is complex real ext-real Element of REAL
(Im (Sum ((b * <i>)))) ^2 is complex real ext-real Element of REAL
(Im (Sum ((b * <i>)))) * (Im (Sum ((b * <i>)))) is complex real ext-real set
((Re (Sum ((b * <i>)))) ^2) + ((Im (Sum ((b * <i>)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Sum ((b * <i>)))) ^2) + ((Im (Sum ((b * <i>)))) ^2)) is complex real ext-real Element of REAL
|.(Sum (((th2 . c₁₂) * <i>))).| * |.(Sum ((b * <i>))).| is complex real ext-real Element of REAL
|.(Sum (((th2 . c₁₂) * <i>))).| * 1 is complex real ext-real Element of REAL
(th2 . c₁₂) - 0 is complex real ext-real Element of REAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
(th2 . c₁₂) + (- 0) is complex real ext-real set
abs ((th2 . c₁₂) - 0) is complex real ext-real Element of REAL
Re ((th2 . c₁₂) - 0) is complex real ext-real Element of REAL
(Re ((th2 . c₁₂) - 0)) ^2 is complex real ext-real Element of REAL
(Re ((th2 . c₁₂) - 0)) * (Re ((th2 . c₁₂) - 0)) is complex real ext-real set
Im ((th2 . c₁₂) - 0) is complex real ext-real Element of REAL
(Im ((th2 . c₁₂) - 0)) ^2 is complex real ext-real Element of REAL
(Im ((th2 . c₁₂) - 0)) * (Im ((th2 . c₁₂) - 0)) is complex real ext-real set
((Re ((th2 . c₁₂) - 0)) ^2) + ((Im ((th2 . c₁₂) - 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((th2 . c₁₂) - 0)) ^2) + ((Im ((th2 . c₁₂) - 0)) ^2)) is complex real ext-real Element of REAL
0 + ((th2 . c₁₂) * <i>) is complex set
Re ((th2 . c₁₂) * <i>) is complex real ext-real Element of REAL
Im ((th2 . c₁₂) * <i>) is complex real ext-real Element of REAL
|.((th2 . c₁₂) * <i>).| is complex real ext-real Element of REAL
(Re ((th2 . c₁₂) * <i>)) ^2 is complex real ext-real Element of REAL
(Re ((th2 . c₁₂) * <i>)) * (Re ((th2 . c₁₂) * <i>)) is complex real ext-real set
(Im ((th2 . c₁₂) * <i>)) ^2 is complex real ext-real Element of REAL
(Im ((th2 . c₁₂) * <i>)) * (Im ((th2 . c₁₂) * <i>)) is complex real ext-real set
((Re ((th2 . c₁₂) * <i>)) ^2) + ((Im ((th2 . c₁₂) * <i>)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((th2 . c₁₂) * <i>)) ^2) + ((Im ((th2 . c₁₂) * <i>)) ^2)) is complex real ext-real Element of REAL
abs (th2 . c₁₂) is complex real ext-real Element of REAL
Re (th2 . c₁₂) is complex real ext-real Element of REAL
(Re (th2 . c₁₂)) ^2 is complex real ext-real Element of REAL
(Re (th2 . c₁₂)) * (Re (th2 . c₁₂)) is complex real ext-real set
Im (th2 . c₁₂) is complex real ext-real Element of REAL
(Im (th2 . c₁₂)) ^2 is complex real ext-real Element of REAL
(Im (th2 . c₁₂)) * (Im (th2 . c₁₂)) is complex real ext-real set
((Re (th2 . c₁₂)) ^2) + ((Im (th2 . c₁₂)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (th2 . c₁₂)) ^2) + ((Im (th2 . c₁₂)) ^2)) is complex real ext-real Element of REAL
c₁₂ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₈ . c₁₂ is complex Element of COMPLEX
(c₈ . c₁₂) - 0c is complex Element of COMPLEX
(c₈ . c₁₂) + (- 0c) is complex set
|.((c₈ . c₁₂) - 0c).| is complex real ext-real Element of REAL
Re ((c₈ . c₁₂) - 0c) is complex real ext-real Element of REAL
(Re ((c₈ . c₁₂) - 0c)) ^2 is complex real ext-real Element of REAL
(Re ((c₈ . c₁₂) - 0c)) * (Re ((c₈ . c₁₂) - 0c)) is complex real ext-real set
Im ((c₈ . c₁₂) - 0c) is complex real ext-real Element of REAL
(Im ((c₈ . c₁₂) - 0c)) ^2 is complex real ext-real Element of REAL
(Im ((c₈ . c₁₂) - 0c)) * (Im ((c₈ . c₁₂) - 0c)) is complex real ext-real set
((Re ((c₈ . c₁₂) - 0c)) ^2) + ((Im ((c₈ . c₁₂) - 0c)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((c₈ . c₁₂) - 0c)) ^2) + ((Im ((c₈ . c₁₂) - 0c)) ^2)) is complex real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
lim c₈ is complex Element of COMPLEX
Re c₈ is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
p is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Re c₈) . p is complex real ext-real Element of REAL
PL . p is complex real ext-real Element of REAL
c₈ . p is complex Element of COMPLEX
Re (c₈ . p) is complex real ext-real Element of REAL
th2 . p is complex real ext-real Element of REAL
(th2 . p) * <i> is complex set
(((th2 . p) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((th2 . p) * <i>)) is complex Element of COMPLEX
Partial_Sums (((th2 . p) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((th2 . p) * <i>))) is complex Element of COMPLEX
(Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
Re ((Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
PL . p is complex real ext-real Element of REAL
((th2 ") (#) (th1 /* th2)) . p is complex real ext-real Element of REAL
th2 . p is complex real ext-real Element of REAL
(th2 . p) * <i> is complex set
(((th2 . p) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((th2 . p) * <i>)) is complex Element of COMPLEX
Partial_Sums (((th2 . p) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((th2 . p) * <i>))) is complex Element of COMPLEX
(Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
Re ((Sum (((th2 . p) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
PL is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
c₈ is complex real ext-real set
PL . c₈ is complex real ext-real Element of REAL
c₈ * (() . b) is complex real ext-real Element of REAL
c₈ is complex real ext-real Element of REAL
PL . c₈ is complex real ext-real Element of REAL
(() . b) * c₈ is complex real ext-real Element of REAL
p is complex real ext-real Element of REAL
c₈ is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued linear Element of K19(K20(REAL,REAL))
th2 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued RestFunc-like Element of K19(K20(REAL,REAL))
b - 1 is complex real ext-real Element of REAL
b + (- 1) is complex real ext-real set
b + 1 is complex real ext-real Element of REAL
].(b - 1),(b + 1).[ is open V46() V47() V48() Element of K19(REAL)
p is complex real ext-real Element of REAL
() . p is complex real ext-real Element of REAL
(() . p) - (() . b) is complex real ext-real Element of REAL
- (() . b) is complex real ext-real set
(() . p) + (- (() . b)) is complex real ext-real set
p - b is complex real ext-real Element of REAL
- b is complex real ext-real set
p + (- b) is complex real ext-real set
c₈ . (p - b) is complex real ext-real Element of REAL
th2 . (p - b) is complex real ext-real Element of REAL
(c₈ . (p - b)) + (th2 . (p - b)) is complex real ext-real Element of REAL
b + (p - b) is complex real ext-real Element of REAL
(p - b) * (() . b) is complex real ext-real Element of REAL
(p - b) * <i> is complex set
(((p - b) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum (((p - b) * <i>)) is complex Element of COMPLEX
Partial_Sums (((p - b) * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((p - b) * <i>))) is complex Element of COMPLEX
(Sum (((p - b) * <i>))) * ((() . b) + ((() . b) * <i>)) is complex set
Re ((Sum (((p - b) * <i>))) * ((() . b) + ((() . b) * <i>))) is complex real ext-real Element of REAL
(p - b) * (Re ((Sum (((p - b) * <i>))) * ((() . b) + ((() . b) * <i>)))) is complex real ext-real Element of REAL
((p - b) * (() . b)) + ((p - b) * (Re ((Sum (((p - b) * <i>))) * ((() . b) + ((() . b) * <i>))))) is complex real ext-real Element of REAL
th1 . (p - b) is complex real ext-real Element of REAL
((p - b) * (() . b)) + (th1 . (p - b)) is complex real ext-real Element of REAL
p is complex real ext-real Element of REAL
() . p is complex real ext-real Element of REAL
(() . p) - (() . b) is complex real ext-real Element of REAL
- (() . b) is complex real ext-real set
(() . p) + (- (() . b)) is complex real ext-real set
p - b is complex real ext-real Element of REAL
- b is complex real ext-real set
p + (- b) is complex real ext-real set
c₈ . (p - b) is complex real ext-real Element of REAL
th2 . (p - b) is complex real ext-real Element of REAL
(c₈ . (p - b)) + (th2 . (p - b)) is complex real ext-real Element of REAL
p is open V46() V47() V48() Neighbourhood of b
c₈ . 1 is complex real ext-real Element of REAL
1 * (() . b) is complex real ext-real Element of REAL
p is open V46() V47() V48() Neighbourhood of b
a is complex real ext-real set
diff ((),a) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
b is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
th1 is Relation-like non-empty NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued bounded convergent 0 -convergent bounded_above bounded_below Element of K19(K20(NAT,REAL))
th1 " is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
b /* th1 is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(th1 ") (#) (b /* th1) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((th1 ") (#) (b /* th1)) is complex real ext-real Element of REAL
th2 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((th1 ") (#) (b /* th1)) . th2 is complex real ext-real Element of REAL
th1 . th2 is complex real ext-real Element of REAL
((th1 . th2)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum ((th1 . th2)) is complex Element of COMPLEX
Partial_Sums ((th1 . th2)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((th1 . th2))) is complex Element of COMPLEX
Re (Sum ((th1 . th2))) is complex real ext-real Element of REAL
(() . a) * (Re (Sum ((th1 . th2)))) is complex real ext-real Element of REAL
(th1 ") . th2 is complex real ext-real Element of REAL
(b /* th1) . th2 is complex real ext-real Element of REAL
((th1 ") . th2) * ((b /* th1) . th2) is complex real ext-real Element of REAL
(th1 . th2) " is complex real ext-real Element of REAL
((th1 . th2) ") * ((b /* th1) . th2) is complex real ext-real Element of REAL
dom b is V46() V47() V48() Element of K19(REAL)
rng th1 is V46() V47() V48() Element of K19(REAL)
b . (th1 . th2) is complex real ext-real Element of REAL
((th1 . th2) ") * (b . (th1 . th2)) is complex real ext-real Element of REAL
(th1 . th2) * (() . a) is complex real ext-real Element of REAL
((th1 . th2) * (() . a)) * (Re (Sum ((th1 . th2)))) is complex real ext-real Element of REAL
((th1 . th2) ") * (((th1 . th2) * (() . a)) * (Re (Sum ((th1 . th2))))) is complex real ext-real Element of REAL
((th1 . th2) ") * (th1 . th2) is complex real ext-real Element of REAL
(((th1 . th2) ") * (th1 . th2)) * ((() . a) * (Re (Sum ((th1 . th2))))) is complex real ext-real Element of REAL
1 * ((() . a) * (Re (Sum ((th1 . th2))))) is complex real ext-real Element of REAL
th2 is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
PL is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
c₈ is complex real ext-real Element of REAL
c₈ / (() . a) is complex real ext-real Element of REAL
(() . a) " is complex real ext-real set
c₈ * ((() . a) ") is complex real ext-real set
p is complex real ext-real Element of REAL
lim th1 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
PL . c₁₁ is complex Element of COMPLEX
(PL . c₁₁) - 0c is complex Element of COMPLEX
- 0c is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
(PL . c₁₁) + (- 0c) is complex set
|.((PL . c₁₁) - 0c).| is complex real ext-real Element of REAL
Re ((PL . c₁₁) - 0c) is complex real ext-real Element of REAL
(Re ((PL . c₁₁) - 0c)) ^2 is complex real ext-real Element of REAL
(Re ((PL . c₁₁) - 0c)) * (Re ((PL . c₁₁) - 0c)) is complex real ext-real set
Im ((PL . c₁₁) - 0c) is complex real ext-real Element of REAL
(Im ((PL . c₁₁) - 0c)) ^2 is complex real ext-real Element of REAL
(Im ((PL . c₁₁) - 0c)) * (Im ((PL . c₁₁) - 0c)) is complex real ext-real set
((Re ((PL . c₁₁) - 0c)) ^2) + ((Im ((PL . c₁₁) - 0c)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((PL . c₁₁) - 0c)) ^2) + ((Im ((PL . c₁₁) - 0c)) ^2)) is complex real ext-real Element of REAL
th1 . c₁₁ is complex real ext-real Element of REAL
((th1 . c₁₁)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum ((th1 . c₁₁)) is complex Element of COMPLEX
Partial_Sums ((th1 . c₁₁)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((th1 . c₁₁))) is complex Element of COMPLEX
(Sum ((th1 . c₁₁))) * (() . a) is complex set
|.((Sum ((th1 . c₁₁))) * (() . a)).| is complex real ext-real Element of REAL
Re ((Sum ((th1 . c₁₁))) * (() . a)) is complex real ext-real Element of REAL
(Re ((Sum ((th1 . c₁₁))) * (() . a))) ^2 is complex real ext-real Element of REAL
(Re ((Sum ((th1 . c₁₁))) * (() . a))) * (Re ((Sum ((th1 . c₁₁))) * (() . a))) is complex real ext-real set
Im ((Sum ((th1 . c₁₁))) * (() . a)) is complex real ext-real Element of REAL
(Im ((Sum ((th1 . c₁₁))) * (() . a))) ^2 is complex real ext-real Element of REAL
(Im ((Sum ((th1 . c₁₁))) * (() . a))) * (Im ((Sum ((th1 . c₁₁))) * (() . a))) is complex real ext-real set
((Re ((Sum ((th1 . c₁₁))) * (() . a))) ^2) + ((Im ((Sum ((th1 . c₁₁))) * (() . a))) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((Sum ((th1 . c₁₁))) * (() . a))) ^2) + ((Im ((Sum ((th1 . c₁₁))) * (() . a))) ^2)) is complex real ext-real Element of REAL
|.(Sum ((th1 . c₁₁))).| is complex real ext-real Element of REAL
Re (Sum ((th1 . c₁₁))) is complex real ext-real Element of REAL
(Re (Sum ((th1 . c₁₁)))) ^2 is complex real ext-real Element of REAL
(Re (Sum ((th1 . c₁₁)))) * (Re (Sum ((th1 . c₁₁)))) is complex real ext-real set
Im (Sum ((th1 . c₁₁))) is complex real ext-real Element of REAL
(Im (Sum ((th1 . c₁₁)))) ^2 is complex real ext-real Element of REAL
(Im (Sum ((th1 . c₁₁)))) * (Im (Sum ((th1 . c₁₁)))) is complex real ext-real set
((Re (Sum ((th1 . c₁₁)))) ^2) + ((Im (Sum ((th1 . c₁₁)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Sum ((th1 . c₁₁)))) ^2) + ((Im (Sum ((th1 . c₁₁)))) ^2)) is complex real ext-real Element of REAL
|.(() . a).| is complex real ext-real Element of REAL
Re (() . a) is complex real ext-real Element of REAL
(Re (() . a)) ^2 is complex real ext-real Element of REAL
(Re (() . a)) * (Re (() . a)) is complex real ext-real set
Im (() . a) is complex real ext-real Element of REAL
(Im (() . a)) ^2 is complex real ext-real Element of REAL
(Im (() . a)) * (Im (() . a)) is complex real ext-real set
((Re (() . a)) ^2) + ((Im (() . a)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (() . a)) ^2) + ((Im (() . a)) ^2)) is complex real ext-real Element of REAL
|.(Sum ((th1 . c₁₁))).| * |.(() . a).| is complex real ext-real Element of REAL
|.(Sum ((th1 . c₁₁))).| * (sqrt (((Re (() . a)) ^2) + ((Im (() . a)) ^2))) is complex real ext-real Element of REAL
(() . a) + (0 * <i>) is complex set
|.(Sum ((th1 . c₁₁))).| * (() . a) is complex real ext-real Element of REAL
(th1 . c₁₁) - 0 is complex real ext-real Element of REAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
(th1 . c₁₁) + (- 0) is complex real ext-real set
abs ((th1 . c₁₁) - 0) is complex real ext-real Element of REAL
Re ((th1 . c₁₁) - 0) is complex real ext-real Element of REAL
(Re ((th1 . c₁₁) - 0)) ^2 is complex real ext-real Element of REAL
(Re ((th1 . c₁₁) - 0)) * (Re ((th1 . c₁₁) - 0)) is complex real ext-real set
Im ((th1 . c₁₁) - 0) is complex real ext-real Element of REAL
(Im ((th1 . c₁₁) - 0)) ^2 is complex real ext-real Element of REAL
(Im ((th1 . c₁₁) - 0)) * (Im ((th1 . c₁₁) - 0)) is complex real ext-real set
((Re ((th1 . c₁₁) - 0)) ^2) + ((Im ((th1 . c₁₁) - 0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((th1 . c₁₁) - 0)) ^2) + ((Im ((th1 . c₁₁) - 0)) ^2)) is complex real ext-real Element of REAL
(c₈ / (() . a)) * (() . a) is complex real ext-real Element of REAL
c₁₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
PL . c₁₁ is complex Element of COMPLEX
(PL . c₁₁) - 0c is complex Element of COMPLEX
(PL . c₁₁) + (- 0c) is complex set
|.((PL . c₁₁) - 0c).| is complex real ext-real Element of REAL
Re ((PL . c₁₁) - 0c) is complex real ext-real Element of REAL
(Re ((PL . c₁₁) - 0c)) ^2 is complex real ext-real Element of REAL
(Re ((PL . c₁₁) - 0c)) * (Re ((PL . c₁₁) - 0c)) is complex real ext-real set
Im ((PL . c₁₁) - 0c) is complex real ext-real Element of REAL
(Im ((PL . c₁₁) - 0c)) ^2 is complex real ext-real Element of REAL
(Im ((PL . c₁₁) - 0c)) * (Im ((PL . c₁₁) - 0c)) is complex real ext-real set
((Re ((PL . c₁₁) - 0c)) ^2) + ((Im ((PL . c₁₁) - 0c)) ^2) is complex real ext-real Element of REAL
sqrt (((Re ((PL . c₁₁) - 0c)) ^2) + ((Im ((PL . c₁₁) - 0c)) ^2)) is complex real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
lim PL is complex Element of COMPLEX
Re PL is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
c₈ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Re PL) . c₈ is complex real ext-real Element of REAL
th1 . c₈ is complex real ext-real Element of REAL
((th1 . c₈)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum ((th1 . c₈)) is complex Element of COMPLEX
Partial_Sums ((th1 . c₈)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((th1 . c₈))) is complex Element of COMPLEX
Re (Sum ((th1 . c₈))) is complex real ext-real Element of REAL
(() . a) * (Re (Sum ((th1 . c₈)))) is complex real ext-real Element of REAL
PL . c₈ is complex Element of COMPLEX
Re (PL . c₈) is complex real ext-real Element of REAL
(Sum ((th1 . c₈))) * (() . a) is complex set
Re ((Sum ((th1 . c₈))) * (() . a)) is complex real ext-real Element of REAL
c₈ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Re PL) . c₈ is complex real ext-real Element of REAL
th2 . c₈ is complex real ext-real Element of REAL
th1 . c₈ is complex real ext-real Element of REAL
((th1 . c₈)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum ((th1 . c₈)) is complex Element of COMPLEX
Partial_Sums ((th1 . c₈)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((th1 . c₈))) is complex Element of COMPLEX
Re (Sum ((th1 . c₈))) is complex real ext-real Element of REAL
(() . a) * (Re (Sum ((th1 . c₈)))) is complex real ext-real Element of REAL
c₈ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
th2 . c₈ is complex real ext-real Element of REAL
((th1 ") (#) (b /* th1)) . c₈ is complex real ext-real Element of REAL
th1 . c₈ is complex real ext-real Element of REAL
((th1 . c₈)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum ((th1 . c₈)) is complex Element of COMPLEX
Partial_Sums ((th1 . c₈)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((th1 . c₈))) is complex Element of COMPLEX
Re (Sum ((th1 . c₈))) is complex real ext-real Element of REAL
(() . a) * (Re (Sum ((th1 . c₈)))) is complex real ext-real Element of REAL
th2 is Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
PL is complex real ext-real set
th2 . PL is complex real ext-real Element of REAL
PL * (() . a) is complex real ext-real Element of REAL
PL is complex real ext-real Element of REAL
th2 . PL is complex real ext-real Element of REAL
(() . a) * PL is complex real ext-real Element of REAL
c₈ is complex real ext-real Element of REAL
PL is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued linear Element of K19(K20(REAL,REAL))
th1 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued RestFunc-like Element of K19(K20(REAL,REAL))
a - 1 is complex real ext-real Element of REAL
a + (- 1) is complex real ext-real set
a + 1 is complex real ext-real Element of REAL
].(a - 1),(a + 1).[ is open V46() V47() V48() Element of K19(REAL)
c₈ is complex real ext-real Element of REAL
() . c₈ is complex real ext-real Element of REAL
(() . c₈) - (() . a) is complex real ext-real Element of REAL
- (() . a) is complex real ext-real set
(() . c₈) + (- (() . a)) is complex real ext-real set
c₈ - a is complex real ext-real Element of REAL
- a is complex real ext-real set
c₈ + (- a) is complex real ext-real set
PL . (c₈ - a) is complex real ext-real Element of REAL
th1 . (c₈ - a) is complex real ext-real Element of REAL
(PL . (c₈ - a)) + (th1 . (c₈ - a)) is complex real ext-real Element of REAL
p is complex real ext-real Element of REAL
c₈ - p is complex real ext-real Element of REAL
- p is complex real ext-real set
c₈ + (- p) is complex real ext-real set
p + (c₈ - p) is complex real ext-real Element of REAL
() . p is complex real ext-real Element of REAL
(() . c₈) - (() . p) is complex real ext-real Element of REAL
- (() . p) is complex real ext-real set
(() . c₈) + (- (() . p)) is complex real ext-real set
(c₈ - p) * (() . p) is complex real ext-real Element of REAL
((c₈ - p)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
Sum ((c₈ - p)) is complex Element of COMPLEX
Partial_Sums ((c₈ - p)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((c₈ - p))) is complex Element of COMPLEX
Re (Sum ((c₈ - p))) is complex real ext-real Element of REAL
((c₈ - p) * (() . p)) * (Re (Sum ((c₈ - p)))) is complex real ext-real Element of REAL
((c₈ - p) * (() . p)) + (((c₈ - p) * (() . p)) * (Re (Sum ((c₈ - p))))) is complex real ext-real Element of REAL
b . (c₈ - p) is complex real ext-real Element of REAL
((c₈ - p) * (() . p)) + (b . (c₈ - p)) is complex real ext-real Element of REAL
PL . (c₈ - p) is complex real ext-real Element of REAL
th1 . (c₈ - p) is complex real ext-real Element of REAL
(PL . (c₈ - p)) + (th1 . (c₈ - p)) is complex real ext-real Element of REAL
c₈ is complex real ext-real Element of REAL
() . c₈ is complex real ext-real Element of REAL
(() . c₈) - (() . a) is complex real ext-real Element of REAL
- (() . a) is complex real ext-real set
(() . c₈) + (- (() . a)) is complex real ext-real set
c₈ - a is complex real ext-real Element of REAL
- a is complex real ext-real set
c₈ + (- a) is complex real ext-real set
PL . (c₈ - a) is complex real ext-real Element of REAL
th1 . (c₈ - a) is complex real ext-real Element of REAL
(PL . (c₈ - a)) + (th1 . (c₈ - a)) is complex real ext-real Element of REAL
c₈ is open V46() V47() V48() Neighbourhood of a
PL . 1 is complex real ext-real Element of REAL
1 * (() . a) is complex real ext-real Element of REAL
c₈ is open V46() V47() V48() Neighbourhood of a
a is complex real ext-real set
diff ((),a) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
[#] REAL is closed open V46() V47() V48() Element of K19(REAL)
b is complex real ext-real Element of REAL
a is complex real ext-real set
diff ((),a) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
- (() . a) is complex real ext-real Element of REAL
[#] REAL is closed open V46() V47() V48() Element of K19(REAL)
b is complex real ext-real Element of REAL
a is complex real ext-real set
diff ((),a) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
[#] REAL is closed open V46() V47() V48() Element of K19(REAL)
b is complex real ext-real Element of REAL
[.0,1.] is closed V46() V47() V48() Element of K19(REAL)
1 / 2 is non empty complex real ext-real positive non negative Element of REAL
2 " is non empty complex real ext-real positive non negative set
1 * (2 ") is non empty complex real ext-real positive non negative set
a is complex real ext-real set
() . a is complex real ext-real Element of REAL
(a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Partial_Sums (a) is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
Sum (a) is complex real ext-real Element of REAL
bq is Relation-like NAT -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K19(K20(NAT,NAT))
(Partial_Sums (a)) * bq is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued subsequence of Partial_Sums (a)
lim ((Partial_Sums (a)) * bq) is complex real ext-real Element of REAL
lim (Partial_Sums (a)) is complex real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums (a)) * bq) . b is complex real ext-real Element of REAL
((Partial_Sums (a)) * bq) . 0 is complex real ext-real Element of REAL
bq . 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (a)) . (bq . 0) is complex real ext-real Element of REAL
2 * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() V62() V63() V98() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (a)) . ((2 * 0) + 1) is complex real ext-real Element of REAL
(Partial_Sums (a)) . 0 is complex real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(a) . (0 + 1) is complex real ext-real Element of REAL
((Partial_Sums (a)) . 0) + ((a) . (0 + 1)) is complex real ext-real Element of REAL
(a) . 0 is complex real ext-real Element of REAL
((a) . 0) + ((a) . (0 + 1)) is complex real ext-real Element of REAL
(- 1) |^ 0 is complex real ext-real Element of REAL
(- 1) GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
((- 1) GeoSeq) . 0 is complex Element of COMPLEX
a |^ (2 * 0) is complex real ext-real set
a GeoSeq is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
(a GeoSeq) . (2 * 0) is complex Element of COMPLEX
((- 1) |^ 0) * (a |^ (2 * 0)) is complex real ext-real Element of REAL
(2 * 0) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * 0) is complex real ext-real Element of REAL
(((- 1) |^ 0) * (a |^ (2 * 0))) / ((2 * 0) !) is complex real ext-real Element of REAL
((2 * 0) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ 0) * (a |^ (2 * 0))) * (((2 * 0) !) ") is complex real ext-real set
(a) . 1 is complex real ext-real Element of REAL
((((- 1) |^ 0) * (a |^ (2 * 0))) / ((2 * 0) !)) + ((a) . 1) is complex real ext-real Element of REAL
(- 1) |^ 1 is complex real ext-real Element of REAL
((- 1) GeoSeq) . 1 is complex Element of COMPLEX
2 * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
a |^ (2 * 1) is complex real ext-real set
(a GeoSeq) . (2 * 1) is complex Element of COMPLEX
((- 1) |^ 1) * (a |^ (2 * 1)) is complex real ext-real Element of REAL
(2 * 1) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * 1) is complex real ext-real Element of REAL
(((- 1) |^ 1) * (a |^ (2 * 1))) / ((2 * 1) !) is complex real ext-real Element of REAL
((2 * 1) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ 1) * (a |^ (2 * 1))) * (((2 * 1) !) ") is complex real ext-real set
((((- 1) |^ 0) * (a |^ (2 * 0))) / ((2 * 0) !)) + ((((- 1) |^ 1) * (a |^ (2 * 1))) / ((2 * 1) !)) is complex real ext-real Element of REAL
1 * (a |^ (2 * 0)) is complex real ext-real Element of REAL
(1 * (a |^ (2 * 0))) / ((2 * 0) !) is complex real ext-real Element of REAL
(1 * (a |^ (2 * 0))) * (((2 * 0) !) ") is complex real ext-real set
((1 * (a |^ (2 * 0))) / ((2 * 0) !)) + ((((- 1) |^ 1) * (a |^ (2 * 1))) / ((2 * 1) !)) is complex real ext-real Element of REAL
1 / 1 is non empty complex real ext-real positive non negative Element of REAL
1 " is non empty complex real ext-real positive non negative set
1 * (1 ") is non empty complex real ext-real positive non negative set
(1 / 1) + ((((- 1) |^ 1) * (a |^ (2 * 1))) / ((2 * 1) !)) is complex real ext-real Element of REAL
(- 1) * (a |^ (2 * 1)) is complex real ext-real Element of REAL
((- 1) * (a |^ (2 * 1))) / ((2 * 1) !) is complex real ext-real Element of REAL
((- 1) * (a |^ (2 * 1))) * (((2 * 1) !) ") is complex real ext-real set
1 + (((- 1) * (a |^ (2 * 1))) / ((2 * 1) !)) is complex real ext-real Element of REAL
a * a is complex real ext-real set
(- 1) * (a * a) is complex real ext-real Element of REAL
((- 1) * (a * a)) / ((2 * 1) !) is complex real ext-real Element of REAL
((- 1) * (a * a)) * (((2 * 1) !) ") is complex real ext-real set
1 + (((- 1) * (a * a)) / ((2 * 1) !)) is complex real ext-real Element of REAL
a ^2 is complex real ext-real set
(a ^2) / 2 is complex real ext-real Element of REAL
(a ^2) * (2 ") is complex real ext-real set
1 - ((a ^2) / 2) is complex real ext-real Element of REAL
- ((a ^2) / 2) is complex real ext-real set
1 + (- ((a ^2) / 2)) is complex real ext-real set
1 ^2 is complex real ext-real Element of REAL
1 * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative set
1 - (1 / 2) is complex real ext-real Element of REAL
- (1 / 2) is non empty complex real ext-real non positive negative set
1 + (- (1 / 2)) is complex real ext-real set
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums (a)) * bq) . th1 is complex real ext-real Element of REAL
th1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
((Partial_Sums (a)) * bq) . (th1 + 1) is complex real ext-real Element of REAL
bq . (th1 + 1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (a)) . (bq . (th1 + 1)) is complex real ext-real Element of REAL
2 * (th1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * (th1 + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (a)) . ((2 * (th1 + 1)) + 1) is complex real ext-real Element of REAL
2 * th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(2 * th1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
((2 * th1) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
(Partial_Sums (a)) . (((2 * th1) + 1) + 1) is complex real ext-real Element of REAL
(a) . ((2 * (th1 + 1)) + 1) is complex real ext-real Element of REAL
((Partial_Sums (a)) . (((2 * th1) + 1) + 1)) + ((a) . ((2 * (th1 + 1)) + 1)) is complex real ext-real Element of REAL
(Partial_Sums (a)) . ((2 * th1) + 1) is complex real ext-real Element of REAL
(a) . (((2 * th1) + 1) + 1) is complex real ext-real Element of REAL
((Partial_Sums (a)) . ((2 * th1) + 1)) + ((a) . (((2 * th1) + 1) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (a)) . ((2 * th1) + 1)) + ((a) . (((2 * th1) + 1) + 1))) + ((a) . ((2 * (th1 + 1)) + 1)) is complex real ext-real Element of REAL
bq . th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
(Partial_Sums (a)) . (bq . th1) is complex real ext-real Element of REAL
((Partial_Sums (a)) . (bq . th1)) + ((a) . (((2 * th1) + 1) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (a)) . (bq . th1)) + ((a) . (((2 * th1) + 1) + 1))) + ((a) . ((2 * (th1 + 1)) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (a)) * bq) . th1) + ((a) . (((2 * th1) + 1) + 1)) is complex real ext-real Element of REAL
((((Partial_Sums (a)) * bq) . th1) + ((a) . (((2 * th1) + 1) + 1))) + ((a) . ((2 * (th1 + 1)) + 1)) is complex real ext-real Element of REAL
(((Partial_Sums (a)) * bq) . (th1 + 1)) - (((Partial_Sums (a)) * bq) . th1) is complex real ext-real Element of REAL
- (((Partial_Sums (a)) * bq) . th1) is complex real ext-real set
(((Partial_Sums (a)) * bq) . (th1 + 1)) + (- (((Partial_Sums (a)) * bq) . th1)) is complex real ext-real set
((a) . (((2 * th1) + 1) + 1)) + ((a) . ((2 * (th1 + 1)) + 1)) is complex real ext-real Element of REAL
(- 1) |^ (2 * (th1 + 1)) is complex real ext-real Element of REAL
((- 1) GeoSeq) . (2 * (th1 + 1)) is complex Element of COMPLEX
2 * (2 * (th1 + 1)) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
a |^ (2 * (2 * (th1 + 1))) is complex real ext-real set
(a GeoSeq) . (2 * (2 * (th1 + 1))) is complex Element of COMPLEX
((- 1) |^ (2 * (th1 + 1))) * (a |^ (2 * (2 * (th1 + 1)))) is complex real ext-real Element of REAL
(2 * (2 * (th1 + 1))) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * (2 * (th1 + 1))) is complex real ext-real Element of REAL
(((- 1) |^ (2 * (th1 + 1))) * (a |^ (2 * (2 * (th1 + 1))))) / ((2 * (2 * (th1 + 1))) !) is complex real ext-real Element of REAL
((2 * (2 * (th1 + 1))) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ (2 * (th1 + 1))) * (a |^ (2 * (2 * (th1 + 1))))) * (((2 * (2 * (th1 + 1))) !) ") is complex real ext-real set
1 * (a |^ (2 * (2 * (th1 + 1)))) is complex real ext-real Element of REAL
(1 * (a |^ (2 * (2 * (th1 + 1))))) / ((2 * (2 * (th1 + 1))) !) is complex real ext-real Element of REAL
(1 * (a |^ (2 * (2 * (th1 + 1))))) * (((2 * (2 * (th1 + 1))) !) ") is complex real ext-real set
(a |^ (2 * (2 * (th1 + 1)))) / ((2 * (2 * (th1 + 1))) !) is complex real ext-real Element of REAL
(a |^ (2 * (2 * (th1 + 1)))) * (((2 * (2 * (th1 + 1))) !) ") is complex real ext-real set
(- 1) |^ ((2 * (th1 + 1)) + 1) is complex real ext-real Element of REAL
((- 1) GeoSeq) . ((2 * (th1 + 1)) + 1) is complex Element of COMPLEX
2 * ((2 * (th1 + 1)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() V98() Element of NAT
a |^ (2 * ((2 * (th1 + 1)) + 1)) is complex real ext-real set
(a GeoSeq) . (2 * ((2 * (th1 + 1)) + 1)) is complex Element of COMPLEX
((- 1) |^ ((2 * (th1 + 1)) + 1)) * (a |^ (2 * ((2 * (th1 + 1)) + 1))) is complex real ext-real Element of REAL
(2 * ((2 * (th1 + 1)) + 1)) ! is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative Element of REAL
() . (2 * ((2 * (th1 + 1)) + 1)) is complex real ext-real Element of REAL
(((- 1) |^ ((2 * (th1 + 1)) + 1)) * (a |^ (2 * ((2 * (th1 + 1)) + 1)))) / ((2 * ((2 * (th1 + 1)) + 1)) !) is complex real ext-real Element of REAL
((2 * ((2 * (th1 + 1)) + 1)) !) " is non empty complex real ext-real positive non negative set
(((- 1) |^ ((2 * (th1 + 1)) + 1)) * (a |^ (2 * ((2 * (th1 + 1)) + 1)))) * (((2 * ((2 * (th1 + 1)) + 1)) !) ") is complex real ext-real set
(- 1) * (a |^ (2 * ((2 * (th1 + 1)) + 1))) is complex real ext-real Element of REAL
((- 1) * (a |^ (2 * ((2 * (th1 + 1)) + 1)))) / ((2 * ((2 * (th1 + 1)) + 1)) !) is complex real ext-real Element of REAL
((- 1) * (a |^ (2 * ((2 * (th1 + 1)) + 1)))) * (((2 * ((2 * (th1 + 1)) + 1)) !) ") is complex real ext-real set
1 / ((2 * ((2 * (th1 + 1)) + 1)) !) is non empty complex real ext-real positive non negative Element of REAL
1 * (((2 * ((2 * (th1 + 1)) + 1)) !) ") is non empty complex real ext-real positive non negative set
1 / ((2 * (2 * (th1 + 1))) !) is non empty complex real ext-real positive non negative Element of REAL
1 * (((2 * (2 * (th1 + 1))) !) ") is non empty complex real ext-real positive non negative set
(a |^ (2 * ((2 * (th1 + 1)) + 1))) * (1 / ((2 * ((2 * (th1 + 1)) + 1)) !)) is complex real ext-real Element of REAL
(a |^ (2 * (2 * (th1 + 1)))) * (1 / ((2 * (2 * (th1 + 1))) !)) is complex real ext-real Element of REAL
(a |^ (2 * (2 * (th1 + 1)))) * 1 is complex real ext-real Element of REAL
((a |^ (2 * (2 * (th1 + 1)))) * 1) / ((2 * (2 * (th1 + 1))) !) is complex real ext-real Element of REAL
((a |^ (2 * (2 * (th1 + 1)))) * 1) * (((2 * (2 * (th1 + 1))) !) ") is complex real ext-real set
(a |^ (2 * ((2 * (th1 + 1)) + 1))) * 1 is complex real ext-real Element of REAL
((a |^ (2 * ((2 * (th1 + 1)) + 1))) * 1) / ((2 * ((2 * (th1 + 1)) + 1)) !) is complex real ext-real Element of REAL
((a |^ (2 * ((2 * (th1 + 1)) + 1))) * 1) * (((2 * ((2 * (th1 + 1)) + 1)) !) ") is complex real ext-real set
(((a |^ (2 * (2 * (th1 + 1)))) * 1) / ((2 * (2 * (th1 + 1))) !)) - (((a |^ (2 * ((2 * (th1 + 1)) + 1))) * 1) / ((2 * ((2 * (th1 + 1)) + 1)) !)) is complex real ext-real Element of REAL
- (((a |^ (2 * ((2 * (th1 + 1)) + 1))) * 1) / ((2 * ((2 * (th1 + 1)) + 1)) !)) is complex real ext-real set
(((a |^ (2 * (2 * (th1 + 1)))) * 1) / ((2 * (2 * (th1 + 1))) !)) + (- (((a |^ (2 * ((2 * (th1 + 1)) + 1))) * 1) / ((2 * ((2 * (th1 + 1)) + 1)) !))) is complex real ext-real set
() / () is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
() / () is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
() is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
() is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom () is V46() V47() V48() Element of K19(REAL)
].0,1.[ is open V46() V47() V48() Element of K19(REAL)
() " {0} is V46() V47() V48() Element of K19(REAL)
[.0,1.] \ (() " {0}) is V46() V47() V48() Element of K19(REAL)
(dom ()) \ (() " {0}) is V46() V47() V48() Element of K19(REAL)
[.0,1.] /\ (() " {0}) is V46() V47() V48() Element of K19(REAL)
a is set
() . a is complex real ext-real Element of REAL
(dom ()) /\ ((dom ()) \ (() " {0})) is V46() V47() V48() Element of K19(REAL)
() | [.0,1.] is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom (() | [.0,1.]) is V46() V47() V48() Element of K19(REAL)
(dom ()) /\ [.0,1.] is V46() V47() V48() Element of K19(REAL)
a is complex real ext-real set
(() | [.0,1.]) . a is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
a is complex real ext-real set
diff ((),a) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
diff ((),a) is complex real ext-real Element of REAL
(diff ((),a)) * (() . a) is complex real ext-real Element of REAL
diff ((),a) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
(diff ((),a)) * (() . a) is complex real ext-real Element of REAL
((diff ((),a)) * (() . a)) - ((diff ((),a)) * (() . a)) is complex real ext-real Element of REAL
- ((diff ((),a)) * (() . a)) is complex real ext-real set
((diff ((),a)) * (() . a)) + (- ((diff ((),a)) * (() . a))) is complex real ext-real set
(() . a) ^2 is complex real ext-real Element of REAL
(() . a) * (() . a) is complex real ext-real set
(((diff ((),a)) * (() . a)) - ((diff ((),a)) * (() . a))) / ((() . a) ^2) is complex real ext-real Element of REAL
((() . a) ^2) " is complex real ext-real set
(((diff ((),a)) * (() . a)) - ((diff ((),a)) * (() . a))) * (((() . a) ^2) ") is complex real ext-real set
(() . a) * (() . a) is complex real ext-real Element of REAL
((() . a) * (() . a)) - ((diff ((),a)) * (() . a)) is complex real ext-real Element of REAL
((() . a) * (() . a)) + (- ((diff ((),a)) * (() . a))) is complex real ext-real set
(((() . a) * (() . a)) - ((diff ((),a)) * (() . a))) / ((() . a) ^2) is complex real ext-real Element of REAL
(((() . a) * (() . a)) - ((diff ((),a)) * (() . a))) * (((() . a) ^2) ") is complex real ext-real set
- (() . a) is complex real ext-real Element of REAL
(- (() . a)) * (() . a) is complex real ext-real Element of REAL
((() . a) * (() . a)) - ((- (() . a)) * (() . a)) is complex real ext-real Element of REAL
- ((- (() . a)) * (() . a)) is complex real ext-real set
((() . a) * (() . a)) + (- ((- (() . a)) * (() . a))) is complex real ext-real set
(((() . a) * (() . a)) - ((- (() . a)) * (() . a))) / ((() . a) ^2) is complex real ext-real Element of REAL
(((() . a) * (() . a)) - ((- (() . a)) * (() . a))) * (((() . a) ^2) ") is complex real ext-real set
(() . a) * (() . a) is complex real ext-real Element of REAL
((() . a) ^2) + ((() . a) * (() . a)) is complex real ext-real Element of REAL
(((() . a) ^2) + ((() . a) * (() . a))) / ((() . a) ^2) is complex real ext-real Element of REAL
(((() . a) ^2) + ((() . a) * (() . a))) * (((() . a) ^2) ") is complex real ext-real set
1 / ((() . a) ^2) is complex real ext-real Element of REAL
1 * (((() . a) ^2) ") is complex real ext-real set
a is complex real ext-real set
diff ((),a) is complex real ext-real Element of REAL
a is complex real ext-real set
[#] REAL is closed open V46() V47() V48() Element of K19(REAL)
() . a is complex real ext-real Element of REAL
b is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng b is V46() V47() V48() Element of K19(REAL)
lim b is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
() /* b is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim (() /* b) is complex real ext-real Element of REAL
dom b is V46() V47() V48() V49() V50() V51() Element of K19(NAT)
th1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
b . th1 is complex real ext-real Element of REAL
(() | [.0,1.]) . (b . th1) is complex real ext-real Element of REAL
() . (b . th1) is complex real ext-real Element of REAL
(() | [.0,1.]) /* b is Relation-like NAT -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((() | [.0,1.]) /* b) . th1 is complex real ext-real Element of REAL
(() /* b) . th1 is complex real ext-real Element of REAL
(() | [.0,1.]) . a is complex real ext-real Element of REAL
lim ((() | [.0,1.]) /* b) is complex real ext-real Element of REAL
a is complex real ext-real set
() . a is complex real ext-real Element of REAL
b is complex real ext-real set
() . b is complex real ext-real Element of REAL
].a,b.[ is open V46() V47() V48() Element of K19(REAL)
th1 is complex real ext-real Element of REAL
[.a,b.] is closed V46() V47() V48() Element of K19(REAL)
th1 is complex real ext-real Element of REAL
() | [.a,b.] is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
th1 is complex real ext-real Element of REAL
diff ((),th1) is complex real ext-real Element of REAL
th2 is complex real ext-real Element of REAL
diff ((),th2) is complex real ext-real Element of REAL
].b,a.[ is open V46() V47() V48() Element of K19(REAL)
th1 is complex real ext-real Element of REAL
[.b,a.] is closed V46() V47() V48() Element of K19(REAL)
th1 is complex real ext-real Element of REAL
() | [.b,a.] is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
th1 is complex real ext-real Element of REAL
diff ((),th1) is complex real ext-real Element of REAL
th2 is complex real ext-real Element of REAL
diff ((),th2) is complex real ext-real Element of REAL
th1 is complex real ext-real Element of REAL
diff ((),th1) is complex real ext-real Element of REAL
() . 0 is complex real ext-real Element of REAL
() . 1 is complex real ext-real Element of REAL
(() . 0) " is complex real ext-real Element of REAL
(() . 0) * ((() . 0) ") is complex real ext-real Element of REAL
(() . 1) / (() . 1) is complex real ext-real Element of REAL
(() . 1) " is complex real ext-real set
(() . 1) * ((() . 1) ") is complex real ext-real set
4 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
].0,4.[ is open V46() V47() V48() Element of K19(REAL)
[.(() . 0),(() . 1).] is closed V46() V47() V48() Element of K19(REAL)
{ b₁ where b₁ is complex real ext-real Element of REAL : ( () . 0 <= b₁ & b₁ <= () . 1 ) } is set
[.(() . 1),(() . 0).] is closed V46() V47() V48() Element of K19(REAL)
[.(() . 0),(() . 1).] \/ [.(() . 1),(() . 0).] is V46() V47() V48() Element of K19(REAL)
a is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
a * 4 is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
b / 4 is complex real ext-real Element of REAL
4 " is non empty complex real ext-real positive non negative set
b * (4 ") is complex real ext-real set
() . (b / 4) is complex real ext-real Element of REAL
b / 4 is complex real ext-real Element of REAL
() . (b / 4) is complex real ext-real Element of REAL
1 * 4 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
a is complex real ext-real set
a / 4 is complex real ext-real Element of REAL
4 " is non empty complex real ext-real positive non negative set
a * (4 ") is complex real ext-real set
() . (a / 4) is complex real ext-real Element of REAL
b is complex real ext-real set
b / 4 is complex real ext-real Element of REAL
b * (4 ") is complex real ext-real set
() . (b / 4) is complex real ext-real Element of REAL
4 / 4 is non empty complex real ext-real positive non negative Element of REAL
4 * (4 ") is non empty complex real ext-real positive non negative set
() is complex real ext-real set
() is complex real ext-real Element of REAL
() / 4 is complex real ext-real Element of REAL
4 " is non empty complex real ext-real positive non negative set
() * (4 ") is complex real ext-real set
() . (() / 4) is complex real ext-real Element of REAL
() . (() / 4) is complex real ext-real Element of REAL
4 / 4 is non empty complex real ext-real positive non negative Element of REAL
4 * (4 ") is non empty complex real ext-real positive non negative set
() . (() / 4) is complex real ext-real Element of REAL
(() . (() / 4)) " is complex real ext-real Element of REAL
(() . (() / 4)) * ((() . (() / 4)) ") is complex real ext-real Element of REAL
a is complex real ext-real set
() . a is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
b is complex real ext-real set
a + b is complex real ext-real set
() . (a + b) is complex real ext-real Element of REAL
() . b is complex real ext-real Element of REAL
(() . a) * (() . b) is complex real ext-real Element of REAL
() . b is complex real ext-real Element of REAL
(() . a) * (() . b) is complex real ext-real Element of REAL
((() . a) * (() . b)) + ((() . a) * (() . b)) is complex real ext-real Element of REAL
() . (a + b) is complex real ext-real Element of REAL
(() . a) * (() . b) is complex real ext-real Element of REAL
(() . a) * (() . b) is complex real ext-real Element of REAL
((() . a) * (() . b)) - ((() . a) * (() . b)) is complex real ext-real Element of REAL
- ((() . a) * (() . b)) is complex real ext-real set
((() . a) * (() . b)) + (- ((() . a) * (() . b))) is complex real ext-real set
th1 is complex real ext-real Element of REAL
th2 is complex real ext-real Element of REAL
th1 + th2 is complex real ext-real Element of REAL
(th1 + th2) * <i> is complex set
0 + 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() V62() V63() Element of NAT
(0 + 0) + ((th1 + th2) * <i>) is complex set
th1 * <i> is complex set
((th1 * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((th1 * <i>)) is complex Element of COMPLEX
Partial_Sums ((th1 * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((th1 * <i>))) is complex Element of COMPLEX
th2 * <i> is complex set
((th2 * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum ((th2 * <i>)) is complex Element of COMPLEX
Partial_Sums ((th2 * <i>)) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums ((th2 * <i>))) is complex Element of COMPLEX
(Sum ((th1 * <i>))) * (Sum ((th2 * <i>))) is complex Element of COMPLEX
(th1 * <i>) + (th2 * <i>) is complex set
(((th1 * <i>) + (th2 * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent summable absolutely_summable Element of K19(K20(NAT,COMPLEX))
Sum (((th1 * <i>) + (th2 * <i>))) is complex Element of COMPLEX
Partial_Sums (((th1 * <i>) + (th2 * <i>))) is Relation-like NAT -defined COMPLEX -valued Function-like non empty total quasi_total complex-valued bounded convergent Element of K19(K20(NAT,COMPLEX))
lim (Partial_Sums (((th1 * <i>) + (th2 * <i>)))) is complex Element of COMPLEX
() . (th1 + th2) is complex real ext-real Element of REAL
() . (th1 + th2) is complex real ext-real Element of REAL
(() . (th1 + th2)) * <i> is complex set
(() . (th1 + th2)) + ((() . (th1 + th2)) * <i>) is complex set
() . th1 is complex real ext-real Element of REAL
() . th1 is complex real ext-real Element of REAL
(() . th1) * <i> is complex set
(() . th1) + ((() . th1) * <i>) is complex set
((() . th1) + ((() . th1) * <i>)) * (Sum ((th2 * <i>))) is complex set
() . th2 is complex real ext-real Element of REAL
() . th2 is complex real ext-real Element of REAL
(() . th2) * <i> is complex set
(() . th2) + ((() . th2) * <i>) is complex set
((() . th1) + ((() . th1) * <i>)) * ((() . th2) + ((() . th2) * <i>)) is complex set
(() . th1) * (() . th2) is complex real ext-real Element of REAL
(() . th1) * (() . th2) is complex real ext-real Element of REAL
((() . th1) * (() . th2)) - ((() . th1) * (() . th2)) is complex real ext-real Element of REAL
- ((() . th1) * (() . th2)) is complex real ext-real set
((() . th1) * (() . th2)) + (- ((() . th1) * (() . th2))) is complex real ext-real set
(() . th1) * (() . th2) is complex real ext-real Element of REAL
(() . th1) * (() . th2) is complex real ext-real Element of REAL
((() . th1) * (() . th2)) + ((() . th1) * (() . th2)) is complex real ext-real Element of REAL
(((() . th1) * (() . th2)) + ((() . th1) * (() . th2))) * <i> is complex set
(((() . th1) * (() . th2)) - ((() . th1) * (() . th2))) + ((((() . th1) * (() . th2)) + ((() . th1) * (() . th2))) * <i>) is complex set
a is complex real ext-real set
b is complex real ext-real set
a + b is complex real ext-real set
((a + b)) is complex real ext-real set
() . (a + b) is complex real ext-real Element of REAL
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
(b) is complex real ext-real set
() . b is complex real ext-real Element of REAL
(a) * (b) is complex real ext-real set
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
(b) is complex real ext-real set
() . b is complex real ext-real Element of REAL
(a) * (b) is complex real ext-real set
((a) * (b)) + ((a) * (b)) is complex real ext-real set
th1 is complex real ext-real set
th2 is complex real ext-real set
th1 + th2 is complex real ext-real set
((th1 + th2)) is complex real ext-real set
() . (th1 + th2) is complex real ext-real Element of REAL
(th1) is complex real ext-real set
() . th1 is complex real ext-real Element of REAL
(th2) is complex real ext-real set
() . th2 is complex real ext-real Element of REAL
(th1) * (th2) is complex real ext-real set
(th1) is complex real ext-real set
() . th1 is complex real ext-real Element of REAL
(th2) is complex real ext-real set
() . th2 is complex real ext-real Element of REAL
(th1) * (th2) is complex real ext-real set
((th1) * (th2)) - ((th1) * (th2)) is complex real ext-real set
- ((th1) * (th2)) is complex real ext-real set
((th1) * (th2)) + (- ((th1) * (th2))) is complex real ext-real set
() / 2 is complex real ext-real Element of REAL
() * (2 ") is complex real ext-real set
() . (() / 2) is complex real ext-real Element of REAL
() . (() / 2) is complex real ext-real Element of REAL
() . () is complex real ext-real Element of REAL
() . () is complex real ext-real Element of REAL
() + (() / 2) is complex real ext-real Element of REAL
() . (() + (() / 2)) is complex real ext-real Element of REAL
() . (() + (() / 2)) is complex real ext-real Element of REAL
2 * () is complex real ext-real Element of REAL
() . (2 * ()) is complex real ext-real Element of REAL
() . (2 * ()) is complex real ext-real Element of REAL
(() / 4) + (() / 4) is complex real ext-real Element of REAL
() . ((() / 4) + (() / 4)) is complex real ext-real Element of REAL
(() . (() / 4)) * (() . (() / 4)) is complex real ext-real Element of REAL
((() . (() / 4)) * (() . (() / 4))) - ((() . (() / 4)) * (() . (() / 4))) is complex real ext-real Element of REAL
- ((() . (() / 4)) * (() . (() / 4))) is complex real ext-real set
((() . (() / 4)) * (() . (() / 4))) + (- ((() . (() / 4)) * (() . (() / 4)))) is complex real ext-real set
() . ((() / 4) + (() / 4)) is complex real ext-real Element of REAL
(() . (() / 4)) * (() . (() / 4)) is complex real ext-real Element of REAL
((() . (() / 4)) * (() . (() / 4))) + ((() . (() / 4)) * (() . (() / 4))) is complex real ext-real Element of REAL
(() / 2) + (() / 2) is complex real ext-real Element of REAL
() . ((() / 2) + (() / 2)) is complex real ext-real Element of REAL
0 * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() V62() V63() Element of NAT
(() . (() / 2)) * (() . (() / 2)) is complex real ext-real Element of REAL
(0 * 0) - ((() . (() / 2)) * (() . (() / 2))) is complex real ext-real Element of REAL
- ((() . (() / 2)) * (() . (() / 2))) is complex real ext-real set
(0 * 0) + (- ((() . (() / 2)) * (() . (() / 2)))) is complex real ext-real set
() . ((() / 2) + (() / 2)) is complex real ext-real Element of REAL
(() . (() / 2)) * (() . (() / 2)) is complex real ext-real Element of REAL
(() . (() / 2)) * (() . (() / 2)) is complex real ext-real Element of REAL
((() . (() / 2)) * (() . (() / 2))) + ((() . (() / 2)) * (() . (() / 2))) is complex real ext-real Element of REAL
(() . ()) * (() . (() / 2)) is complex real ext-real Element of REAL
(() . ()) * (() . (() / 2)) is complex real ext-real Element of REAL
((() . ()) * (() . (() / 2))) - ((() . ()) * (() . (() / 2))) is complex real ext-real Element of REAL
- ((() . ()) * (() . (() / 2))) is complex real ext-real set
((() . ()) * (() . (() / 2))) + (- ((() . ()) * (() . (() / 2)))) is complex real ext-real set
(() . ()) * (() . (() / 2)) is complex real ext-real Element of REAL
(() . ()) * (() . (() / 2)) is complex real ext-real Element of REAL
((() . ()) * (() . (() / 2))) + ((() . ()) * (() . (() / 2))) is complex real ext-real Element of REAL
() + () is complex real ext-real Element of REAL
() . (() + ()) is complex real ext-real Element of REAL
(- 1) * (- 1) is non empty complex real ext-real positive non negative Element of REAL
(() . ()) * (() . ()) is complex real ext-real Element of REAL
((- 1) * (- 1)) - ((() . ()) * (() . ())) is complex real ext-real Element of REAL
- ((() . ()) * (() . ())) is complex real ext-real set
((- 1) * (- 1)) + (- ((() . ()) * (() . ()))) is complex real ext-real set
() . (() + ()) is complex real ext-real Element of REAL
(() . ()) * (() . ()) is complex real ext-real Element of REAL
(() . ()) * (() . ()) is complex real ext-real Element of REAL
((() . ()) * (() . ())) + ((() . ()) * (() . ())) is complex real ext-real Element of REAL
((() / 2)) is complex real ext-real Element of REAL
((() / 2)) is complex real ext-real Element of REAL
(()) is complex real ext-real Element of REAL
(()) is complex real ext-real Element of REAL
((() + (() / 2))) is complex real ext-real Element of REAL
((() + (() / 2))) is complex real ext-real Element of REAL
((2 * ())) is complex real ext-real Element of REAL
((2 * ())) is complex real ext-real Element of REAL
a is complex real ext-real set
a + (2 * ()) is complex real ext-real Element of REAL
() . (a + (2 * ())) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
() . (a + (2 * ())) is complex real ext-real Element of REAL
() . a is complex real ext-real Element of REAL
(() / 2) - a is complex real ext-real Element of REAL
- a is complex real ext-real set
(() / 2) + (- a) is complex real ext-real set
() . ((() / 2) - a) is complex real ext-real Element of REAL
() . ((() / 2) - a) is complex real ext-real Element of REAL
(() / 2) + a is complex real ext-real Element of REAL
() . ((() / 2) + a) is complex real ext-real Element of REAL
() . ((() / 2) + a) is complex real ext-real Element of REAL
- (() . a) is complex real ext-real Element of REAL
() + a is complex real ext-real Element of REAL
() . (() + a) is complex real ext-real Element of REAL
() . (() + a) is complex real ext-real Element of REAL
- (() . a) is complex real ext-real Element of REAL
(() . a) * 1 is complex real ext-real Element of REAL
(() . a) * 0 is complex real ext-real Element of REAL
((() . a) * 1) + ((() . a) * 0) is complex real ext-real Element of REAL
(() . a) * 1 is complex real ext-real Element of REAL
(() . a) * 0 is complex real ext-real Element of REAL
((() . a) * 1) - ((() . a) * 0) is complex real ext-real Element of REAL
- ((() . a) * 0) is complex real ext-real set
((() . a) * 1) + (- ((() . a) * 0)) is complex real ext-real set
() . (- a) is complex real ext-real Element of REAL
(() . (() / 2)) * (() . (- a)) is complex real ext-real Element of REAL
() . (- a) is complex real ext-real Element of REAL
(() . (() / 2)) * (() . (- a)) is complex real ext-real Element of REAL
((() . (() / 2)) * (() . (- a))) + ((() . (() / 2)) * (() . (- a))) is complex real ext-real Element of REAL
(() . (() / 2)) * (() . (- a)) is complex real ext-real Element of REAL
(() . (() / 2)) * (() . (- a)) is complex real ext-real Element of REAL
((() . (() / 2)) * (() . (- a))) - ((() . (() / 2)) * (() . (- a))) is complex real ext-real Element of REAL
- ((() . (() / 2)) * (() . (- a))) is complex real ext-real set
((() . (() / 2)) * (() . (- a))) + (- ((() . (() / 2)) * (() . (- a)))) is complex real ext-real set
1 * (- (() . a)) is complex real ext-real Element of REAL
0 - (1 * (- (() . a))) is complex real ext-real Element of REAL
- (1 * (- (() . a))) is complex real ext-real set
0 + (- (1 * (- (() . a)))) is complex real ext-real set
1 * (() . a) is complex real ext-real Element of REAL
0 * (() . a) is complex real ext-real Element of REAL
(1 * (() . a)) + (0 * (() . a)) is complex real ext-real Element of REAL
(() . (() / 2)) * (() . a) is complex real ext-real Element of REAL
(() . (() / 2)) * (() . a) is complex real ext-real Element of REAL
((() . (() / 2)) * (() . a)) - ((() . (() / 2)) * (() . a)) is complex real ext-real Element of REAL
- ((() . (() / 2)) * (() . a)) is complex real ext-real set
((() . (() / 2)) * (() . a)) + (- ((() . (() / 2)) * (() . a))) is complex real ext-real set
(() . ()) * (() . a) is complex real ext-real Element of REAL
(() . ()) * (() . a) is complex real ext-real Element of REAL
((() . ()) * (() . a)) + ((() . ()) * (() . a)) is complex real ext-real Element of REAL
(- 1) * (() . a) is complex real ext-real Element of REAL
((- 1) * (() . a)) - (0 * (() . a)) is complex real ext-real Element of REAL
- (0 * (() . a)) is complex real ext-real set
((- 1) * (() . a)) + (- (0 * (() . a))) is complex real ext-real set
a is complex real ext-real set
a + (2 * ()) is complex real ext-real Element of REAL
((a + (2 * ()))) is complex real ext-real Element of REAL
() . (a + (2 * ())) is complex real ext-real Element of REAL
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
b is complex real ext-real set
b + (2 * ()) is complex real ext-real Element of REAL
((b + (2 * ()))) is complex real ext-real Element of REAL
() . (b + (2 * ())) is complex real ext-real Element of REAL
(b) is complex real ext-real set
() . b is complex real ext-real Element of REAL
th1 is complex real ext-real set
(() / 2) - th1 is complex real ext-real Element of REAL
- th1 is complex real ext-real set
(() / 2) + (- th1) is complex real ext-real set
(((() / 2) - th1)) is complex real ext-real Element of REAL
() . ((() / 2) - th1) is complex real ext-real Element of REAL
(th1) is complex real ext-real set
() . th1 is complex real ext-real Element of REAL
th2 is complex real ext-real set
(() / 2) - th2 is complex real ext-real Element of REAL
- th2 is complex real ext-real set
(() / 2) + (- th2) is complex real ext-real set
(((() / 2) - th2)) is complex real ext-real Element of REAL
() . ((() / 2) - th2) is complex real ext-real Element of REAL
(th2) is complex real ext-real set
() . th2 is complex real ext-real Element of REAL
PL is complex real ext-real set
(() / 2) + PL is complex real ext-real Element of REAL
(((() / 2) + PL)) is complex real ext-real Element of REAL
() . ((() / 2) + PL) is complex real ext-real Element of REAL
(PL) is complex real ext-real set
() . PL is complex real ext-real Element of REAL
c₈ is complex real ext-real set
(() / 2) + c₈ is complex real ext-real Element of REAL
(((() / 2) + c₈)) is complex real ext-real Element of REAL
() . ((() / 2) + c₈) is complex real ext-real Element of REAL
(c₈) is complex real ext-real set
() . c₈ is complex real ext-real Element of REAL
- (c₈) is complex real ext-real set
p is complex real ext-real set
() + p is complex real ext-real Element of REAL
((() + p)) is complex real ext-real Element of REAL
() . (() + p) is complex real ext-real Element of REAL
(p) is complex real ext-real set
() . p is complex real ext-real Element of REAL
- (p) is complex real ext-real set
k is complex real ext-real set
() + k is complex real ext-real Element of REAL
((() + k)) is complex real ext-real Element of REAL
() . (() + k) is complex real ext-real Element of REAL
(k) is complex real ext-real set
() . k is complex real ext-real Element of REAL
- (k) is complex real ext-real set
a is complex real ext-real set
() . a is complex real ext-real Element of REAL
() | REAL is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
[.0,a.] is closed V46() V47() V48() Element of K19(REAL)
() | [.0,a.] is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
].0,a.[ is open V46() V47() V48() Element of K19(REAL)
(() . a) - (() . 0) is complex real ext-real Element of REAL
- (() . 0) is complex real ext-real set
(() . a) + (- (() . 0)) is complex real ext-real set
a - 0 is complex real ext-real Element of REAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() set
a + (- 0) is complex real ext-real set
((() . a) - (() . 0)) / (a - 0) is complex real ext-real Element of REAL
(a - 0) " is complex real ext-real set
((() . a) - (() . 0)) * ((a - 0) ") is complex real ext-real set
b is complex real ext-real Element of REAL
diff ((),b) is complex real ext-real Element of REAL
() . b is complex real ext-real Element of REAL
].0,(() / 2).[ is open V46() V47() V48() Element of K19(REAL)
a is complex real ext-real set
() . a is complex real ext-real Element of REAL
() | REAL is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
[.0,a.] is closed V46() V47() V48() Element of K19(REAL)
() | [.0,a.] is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
[.(() . 0),(() . a).] is closed V46() V47() V48() Element of K19(REAL)
[.(() . a),(() . 0).] is closed V46() V47() V48() Element of K19(REAL)
[.(() . 0),(() . a).] \/ [.(() . a),(() . 0).] is V46() V47() V48() Element of K19(REAL)
b is complex real ext-real Element of REAL
() . b is complex real ext-real Element of REAL
b is complex real ext-real Element of REAL
() . b is complex real ext-real Element of REAL
(() / 2) / 2 is complex real ext-real Element of REAL
(() / 2) * (2 ") is complex real ext-real set
b / 2 is complex real ext-real Element of REAL
b * (2 ") is complex real ext-real set
4 / 4 is non empty complex real ext-real positive non negative Element of REAL
4 * (4 ") is non empty complex real ext-real positive non negative set
(b / 2) + (b / 2) is complex real ext-real Element of REAL
() . ((b / 2) + (b / 2)) is complex real ext-real Element of REAL
() . (b / 2) is complex real ext-real Element of REAL
(() . (b / 2)) ^2 is complex real ext-real Element of REAL
(() . (b / 2)) * (() . (b / 2)) is complex real ext-real set
() . (b / 2) is complex real ext-real Element of REAL
(() . (b / 2)) * (() . (b / 2)) is complex real ext-real Element of REAL
((() . (b / 2)) ^2) - ((() . (b / 2)) * (() . (b / 2))) is complex real ext-real Element of REAL
- ((() . (b / 2)) * (() . (b / 2))) is complex real ext-real set
((() . (b / 2)) ^2) + (- ((() . (b / 2)) * (() . (b / 2)))) is complex real ext-real set
(() . (b / 2)) - (() . (b / 2)) is complex real ext-real Element of REAL
- (() . (b / 2)) is complex real ext-real set
(() . (b / 2)) + (- (() . (b / 2))) is complex real ext-real set
(() . (b / 2)) + (() . (b / 2)) is complex real ext-real Element of REAL
((() . (b / 2)) - (() . (b / 2))) * ((() . (b / 2)) + (() . (b / 2))) is complex real ext-real Element of REAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V46() V47() V48() V49() V50() V51() V52() Element of REAL
4 * 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V46() V47() V48() V49() V50() V51() V62() V63() Element of NAT
4 * (b / 2) is complex real ext-real Element of REAL
2 * b is complex real ext-real Element of REAL
(() . (b / 2)) " is complex real ext-real Element of REAL
(() . (b / 2)) * ((() . (b / 2)) ") is complex real ext-real Element of REAL
(2 * b) / 4 is complex real ext-real Element of REAL
(2 * b) * (4 ") is complex real ext-real set
() . ((2 * b) / 4) is complex real ext-real Element of REAL
a is complex real ext-real set
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
a is complex real ext-real set
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
b is complex real ext-real set
a - b is complex real ext-real set
- b is complex real ext-real set
a + (- b) is complex real ext-real set
((a - b)) is complex real ext-real set
() . (a - b) is complex real ext-real Element of REAL
(b) is complex real ext-real set
() . b is complex real ext-real Element of REAL
(a) * (b) is complex real ext-real set
(b) is complex real ext-real set
() . b is complex real ext-real Element of REAL
(a) * (b) is complex real ext-real set
((a) * (b)) - ((a) * (b)) is complex real ext-real set
- ((a) * (b)) is complex real ext-real set
((a) * (b)) + (- ((a) * (b))) is complex real ext-real set
() . (- b) is complex real ext-real Element of REAL
(() . a) * (() . (- b)) is complex real ext-real Element of REAL
() . (- b) is complex real ext-real Element of REAL
(() . a) * (() . (- b)) is complex real ext-real Element of REAL
((() . a) * (() . (- b))) + ((() . a) * (() . (- b))) is complex real ext-real Element of REAL
(() . a) * (() . b) is complex real ext-real Element of REAL
((() . a) * (() . b)) + ((() . a) * (() . (- b))) is complex real ext-real Element of REAL
- (() . b) is complex real ext-real Element of REAL
(() . a) * (- (() . b)) is complex real ext-real Element of REAL
((() . a) * (() . b)) + ((() . a) * (- (() . b))) is complex real ext-real Element of REAL
a is complex real ext-real set
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
(a) is complex real ext-real set
() . a is complex real ext-real Element of REAL
b is complex real ext-real set
a - b is complex real ext-real set
- b is complex real ext-real set
a + (- b) is complex real ext-real set
((a - b)) is complex real ext-real set
() . (a - b) is complex real ext-real Element of REAL
(b) is complex real ext-real set
() . b is complex real ext-real Element of REAL
(a) * (b) is complex real ext-real set
(b) is complex real ext-real set
() . b is complex real ext-real Element of REAL
(a) * (b) is complex real ext-real set
((a) * (b)) + ((a) * (b)) is complex real ext-real set
() . (- b) is complex real ext-real Element of REAL
(() . a) * (() . (- b)) is complex real ext-real Element of REAL
() . (- b) is complex real ext-real Element of REAL
(() . a) * (() . (- b)) is complex real ext-real Element of REAL
((() . a) * (() . (- b))) - ((() . a) * (() . (- b))) is complex real ext-real Element of REAL
- ((() . a) * (() . (- b))) is complex real ext-real set
((() . a) * (() . (- b))) + (- ((() . a) * (() . (- b)))) is complex real ext-real set
(() . a) * (() . b) is complex real ext-real Element of REAL
((() . a) * (() . b)) - ((() . a) * (() . (- b))) is complex real ext-real Element of REAL
((() . a) * (() . b)) + (- ((() . a) * (() . (- b)))) is complex real ext-real set
- (() . b) is complex real ext-real Element of REAL
(() . a) * (- (() . b)) is complex real ext-real Element of REAL
((() . a) * (() . b)) - ((() . a) * (- (() . b))) is complex real ext-real Element of REAL
- ((() . a) * (- (() . b))) is complex real ext-real set
((() . a) * (() . b)) + (- ((() . a) * (- (() . b)))) is complex real ext-real set
() | REAL is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
() | REAL is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
() | REAL is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))