:: UNIROOTS semantic presentation

REAL is non empty non trivial non finite V67() V68() V69() V73() set
NAT is ordinal non empty non trivial non finite cardinal limit_cardinal V67() V68() V69() V70() V71() V72() V73() Element of bool REAL
bool REAL is non empty non trivial non finite set
F_Complex is non empty non degenerated non trivial right_complementable almost_left_invertible strict unital associative commutative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() algebraic-closed domRing-like doubleLoopStr
the carrier of F_Complex is non empty non trivial set
COMPLEX is non empty non trivial non finite V67() V73() set
NAT is ordinal non empty non trivial non finite cardinal limit_cardinal V67() V68() V69() V70() V71() V72() V73() set
bool NAT is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
K227(NAT) is V55() set
RAT is non empty non trivial non finite V67() V68() V69() V70() V73() set
INT is non empty non trivial non finite V67() V68() V69() V70() V71() V73() set
[:REAL,REAL:] is Relation-like non empty non trivial non finite V57() V58() V59() set
bool [:REAL,REAL:] is non empty non trivial non finite set
[:NAT,REAL:] is Relation-like non empty non trivial non finite V57() V58() V59() set
bool [:NAT,REAL:] is non empty non trivial non finite set
[:NAT,COMPLEX:] is Relation-like non empty non trivial non finite V57() set
bool [:NAT,COMPLEX:] is non empty non trivial non finite set
[:COMPLEX,COMPLEX:] is Relation-like non empty non trivial non finite V57() set
bool [:COMPLEX,COMPLEX:] is non empty non trivial non finite set
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered rational V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of NAT
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support set
the Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support set is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support set
1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
2 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
K357(0,1,2) is finite V67() V68() V69() V70() V71() V72() set
[:K357(0,1,2),K357(0,1,2):] is Relation-like RAT -valued INT -valued finite V57() V58() V59() V60() set
[:[:K357(0,1,2),K357(0,1,2):],K357(0,1,2):] is Relation-like RAT -valued INT -valued finite V57() V58() V59() V60() set
bool [:[:K357(0,1,2),K357(0,1,2):],K357(0,1,2):] is finite V44() set
bool [:K357(0,1,2),K357(0,1,2):] is finite V44() set
1 -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of NAT * : len b₁ = 1 } is set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like non empty non trivial non finite V57() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial non finite set
[:[:REAL,REAL:],REAL:] is Relation-like non empty non trivial non finite V57() V58() V59() set
bool [:[:REAL,REAL:],REAL:] is non empty non trivial non finite set
[:RAT,RAT:] is Relation-like RAT -valued non empty non trivial non finite V57() V58() V59() set
bool [:RAT,RAT:] is non empty non trivial non finite set
[:[:RAT,RAT:],RAT:] is Relation-like RAT -valued non empty non trivial non finite V57() V58() V59() set
bool [:[:RAT,RAT:],RAT:] is non empty non trivial non finite set
[:INT,INT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() set
bool [:INT,INT:] is non empty non trivial non finite set
[:[:INT,INT:],INT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() set
bool [:[:INT,INT:],INT:] is non empty non trivial non finite set
[:NAT,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() V60() set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() V60() set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite set
3 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg 1 is non empty trivial finite 1 -element V67() V68() V69() V70() V71() V72() Element of bool NAT
PI is V31() real ext-real Element of REAL
PI / 2 is V31() real ext-real Element of REAL
- (PI / 2) is V31() real ext-real Element of REAL
2 * PI is V31() real ext-real Element of REAL
3 / 2 is non empty V31() real ext-real positive non negative Element of REAL
(3 / 2) * PI is V31() real ext-real Element of REAL
<i> is V31() Element of COMPLEX
Arg 1 is V31() real ext-real Element of REAL
addcomplex is Relation-like [:COMPLEX,COMPLEX:] -defined COMPLEX -valued Function-like V25([:COMPLEX,COMPLEX:], COMPLEX ) commutative associative V57() Element of bool [:[:COMPLEX,COMPLEX:],COMPLEX:]
multcomplex is Relation-like [:COMPLEX,COMPLEX:] -defined COMPLEX -valued Function-like V25([:COMPLEX,COMPLEX:], COMPLEX ) commutative associative V57() Element of bool [:[:COMPLEX,COMPLEX:],COMPLEX:]
1r is V31() Element of COMPLEX
0c is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of COMPLEX
0. F_Complex is V31() zero right_complementable Element of the carrier of F_Complex
the ZeroF of F_Complex is V31() right_complementable Element of the carrier of F_Complex
1_ F_Complex is V31() right_complementable Element of the carrier of F_Complex
1. F_Complex is V31() non zero right_complementable Element of the carrier of F_Complex
the OneF of F_Complex is V31() right_complementable Element of the carrier of F_Complex
|.(1. F_Complex).| is V31() real ext-real Element of REAL
Re (1. F_Complex) is V31() real ext-real Element of REAL
(Re (1. F_Complex)) ^2 is V31() real ext-real Element of REAL
K104((Re (1. F_Complex)),(Re (1. F_Complex))) is V31() real ext-real set
Im (1. F_Complex) is V31() real ext-real Element of REAL
(Im (1. F_Complex)) ^2 is V31() real ext-real Element of REAL
K104((Im (1. F_Complex)),(Im (1. F_Complex))) is V31() real ext-real set
((Re (1. F_Complex)) ^2) + ((Im (1. F_Complex)) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (1. F_Complex)) ^2) + ((Im (1. F_Complex)) ^2)) is V31() real ext-real Element of REAL
power F_Complex is Relation-like [: the carrier of F_Complex,NAT:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex,NAT:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex,NAT:], the carrier of F_Complex:]
[: the carrier of F_Complex,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() V60() set
[:[: the carrier of F_Complex,NAT:], the carrier of F_Complex:] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of F_Complex,NAT:], the carrier of F_Complex:] is non empty non trivial non finite set
|.1r.| is V31() real ext-real Element of REAL
Re 1r is V31() real ext-real Element of REAL
(Re 1r) ^2 is V31() real ext-real Element of REAL
K104((Re 1r),(Re 1r)) is V31() real ext-real set
Im 1r is V31() real ext-real Element of REAL
(Im 1r) ^2 is V31() real ext-real Element of REAL
K104((Im 1r),(Im 1r)) is V31() real ext-real set
((Re 1r) ^2) + ((Im 1r) ^2) is V31() real ext-real Element of REAL
sqrt (((Re 1r) ^2) + ((Im 1r) ^2)) is V31() real ext-real Element of REAL
sin is Relation-like REAL -defined REAL -valued Function-like V25( REAL , REAL ) V57() V58() V59() Element of bool [:REAL,REAL:]
cos is Relation-like REAL -defined REAL -valued Function-like V25( REAL , REAL ) V57() V58() V59() Element of bool [:REAL,REAL:]
K706(cos,0) is V31() real ext-real Element of REAL
K706(sin,0) is V31() real ext-real Element of REAL
cos 0 is V31() real ext-real Element of REAL
sin 0 is V31() real ext-real Element of REAL
cos (PI / 2) is V31() real ext-real Element of REAL
sin (PI / 2) is V31() real ext-real Element of REAL
cos PI is V31() real ext-real Element of REAL
- 1 is non empty V31() real ext-real non positive negative integer Element of REAL
sin PI is V31() real ext-real Element of REAL
PI + (PI / 2) is V31() real ext-real Element of REAL
cos (PI + (PI / 2)) is V31() real ext-real Element of REAL
sin (PI + (PI / 2)) is V31() real ext-real Element of REAL
cos (2 * PI) is V31() real ext-real Element of REAL
sin (2 * PI) is V31() real ext-real Element of REAL
<*> REAL is Relation-like non-empty empty-yielding NAT -defined REAL -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support FinSequence of REAL
Product (<*> REAL) is V31() real ext-real Element of REAL
K105(1) is non empty V31() real ext-real non positive negative integer set
{{}} is functional non empty trivial finite V44() 1 -element V67() V68() V69() V70() V71() V72() set
Z_3 is strict doubleLoopStr
add3 is Relation-like [:K357(0,1,2),K357(0,1,2):] -defined K357(0,1,2) -valued Function-like V25([:K357(0,1,2),K357(0,1,2):],K357(0,1,2)) finite V57() V58() V59() V60() finite-support Element of bool [:[:K357(0,1,2),K357(0,1,2):],K357(0,1,2):]
mult3 is Relation-like [:K357(0,1,2),K357(0,1,2):] -defined K357(0,1,2) -valued Function-like V25([:K357(0,1,2),K357(0,1,2):],K357(0,1,2)) finite V57() V58() V59() V60() finite-support Element of bool [:[:K357(0,1,2),K357(0,1,2):],K357(0,1,2):]
unit3 is ordinal natural V31() real ext-real non negative integer finite cardinal rational Element of K357(0,1,2)
zero3 is ordinal natural V31() real ext-real non negative integer finite cardinal rational Element of K357(0,1,2)
doubleLoopStr(# K357(0,1,2),add3,mult3,unit3,zero3 #) is strict doubleLoopStr
MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal set
n is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined Function-like finite FinSubsequence-like finite-support set
MGFC . 1 is set
<*(MGFC . 1)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
Seg (len MGFC) is finite len MGFC -element V67() V68() V69() V70() V71() V72() Element of bool NAT
dom MGFC is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC | (Seg 1)) . 1 is set
cMGFC is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom cMGFC is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
len cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg cMGFC is finite cMGFC -element V67() V68() V69() V70() V71() V72() Element of bool NAT
MGFC | (Seg cMGFC) is Relation-like NAT -defined Seg cMGFC -defined NAT -defined Function-like finite FinSubsequence-like finite-support set
Seg n is finite n -element V67() V68() V69() V70() V71() V72() Element of bool NAT
(MGFC | (Seg cMGFC)) | (Seg n) is Relation-like NAT -defined Seg n -defined NAT -defined Function-like finite FinSubsequence-like finite-support set
MGFC | (Seg n) is Relation-like NAT -defined Seg n -defined NAT -defined Function-like finite FinSubsequence-like finite-support set
n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom S is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Product S is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) associative Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
K228( the carrier of F_Complex,S, the multF of F_Complex) is V31() right_complementable Element of the carrier of F_Complex
|.(Product S).| is V31() real ext-real Element of REAL
Re (Product S) is V31() real ext-real Element of REAL
(Re (Product S)) ^2 is V31() real ext-real Element of REAL
K104((Re (Product S)),(Re (Product S))) is V31() real ext-real set
Im (Product S) is V31() real ext-real Element of REAL
(Im (Product S)) ^2 is V31() real ext-real Element of REAL
K104((Im (Product S)),(Im (Product S))) is V31() real ext-real set
((Re (Product S)) ^2) + ((Im (Product S)) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (Product S)) ^2) + ((Im (Product S)) ^2)) is V31() real ext-real Element of REAL
fs is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
len fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Product fs is V31() real ext-real Element of REAL
q is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
qc is V31() real ext-real Element of REAL
<*qc*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like V57() V58() V59() V61() V62() V63() V64() finite-support Element of REAL *
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
q ^ <*qc*> is Relation-like NAT -defined REAL -valued Function-like non empty finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
len q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
len <*qc*> is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len q) + (len <*qc*>) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len q) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs . (len S) is V31() real ext-real set
p1 is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
ps is V31() right_complementable Element of the carrier of F_Complex
<*ps*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex *
the carrier of F_Complex * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
p1 ^ <*ps*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len p1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
len <*ps*> is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len p1) + (len <*ps*>) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len p1) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom p1 is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
dom q is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 /. qi is V31() right_complementable Element of the carrier of F_Complex
p1 . qi is set
S . qi is set
S /. qi is V31() right_complementable Element of the carrier of F_Complex
|.(p1 /. qi).| is V31() real ext-real Element of REAL
Re (p1 /. qi) is V31() real ext-real Element of REAL
(Re (p1 /. qi)) ^2 is V31() real ext-real Element of REAL
K104((Re (p1 /. qi)),(Re (p1 /. qi))) is V31() real ext-real set
Im (p1 /. qi) is V31() real ext-real Element of REAL
(Im (p1 /. qi)) ^2 is V31() real ext-real Element of REAL
K104((Im (p1 /. qi)),(Im (p1 /. qi))) is V31() real ext-real set
((Re (p1 /. qi)) ^2) + ((Im (p1 /. qi)) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (p1 /. qi)) ^2) + ((Im (p1 /. qi)) ^2)) is V31() real ext-real Element of REAL
fs . qi is V31() real ext-real set
q . qi is V31() real ext-real set
Product p1 is V31() right_complementable Element of the carrier of F_Complex
K228( the carrier of F_Complex,p1, the multF of F_Complex) is V31() right_complementable Element of the carrier of F_Complex
Product q is V31() real ext-real Element of REAL
(Product q) * qc is V31() real ext-real Element of REAL
S /. (len S) is V31() right_complementable Element of the carrier of F_Complex
S . (len S) is set
i is V31() Element of COMPLEX
|.i.| is V31() real ext-real Element of REAL
Re i is V31() real ext-real Element of REAL
(Re i) ^2 is V31() real ext-real Element of REAL
K104((Re i),(Re i)) is V31() real ext-real set
Im i is V31() real ext-real Element of REAL
(Im i) ^2 is V31() real ext-real Element of REAL
K104((Im i),(Im i)) is V31() real ext-real set
((Re i) ^2) + ((Im i) ^2) is V31() real ext-real Element of REAL
sqrt (((Re i) ^2) + ((Im i) ^2)) is V31() real ext-real Element of REAL
(Product p1) * ps is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((Product p1),ps) is V31() right_complementable Element of the carrier of F_Complex
[(Product p1),ps] is non empty set
{(Product p1),ps} is non empty finite V67() set
{(Product p1)} is non empty trivial finite 1 -element V67() set
{{(Product p1),ps},{(Product p1)}} is non empty finite V44() set
the multF of F_Complex . [(Product p1),ps] is set
(Product p1) * ps is V31() Element of COMPLEX
qi is V31() Element of COMPLEX
|.qi.| is V31() real ext-real Element of REAL
Re qi is V31() real ext-real Element of REAL
(Re qi) ^2 is V31() real ext-real Element of REAL
K104((Re qi),(Re qi)) is V31() real ext-real set
Im qi is V31() real ext-real Element of REAL
(Im qi) ^2 is V31() real ext-real Element of REAL
K104((Im qi),(Im qi)) is V31() real ext-real set
((Re qi) ^2) + ((Im qi) ^2) is V31() real ext-real Element of REAL
sqrt (((Re qi) ^2) + ((Im qi) ^2)) is V31() real ext-real Element of REAL
|.qi.| * |.i.| is V31() real ext-real Element of REAL
n is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom n is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Product n is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) associative Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
K228( the carrier of F_Complex,n, the multF of F_Complex) is V31() right_complementable Element of the carrier of F_Complex
|.(Product n).| is V31() real ext-real Element of REAL
Re (Product n) is V31() real ext-real Element of REAL
(Re (Product n)) ^2 is V31() real ext-real Element of REAL
K104((Re (Product n)),(Re (Product n))) is V31() real ext-real set
Im (Product n) is V31() real ext-real Element of REAL
(Im (Product n)) ^2 is V31() real ext-real Element of REAL
K104((Im (Product n)),(Im (Product n))) is V31() real ext-real set
((Re (Product n)) ^2) + ((Im (Product n)) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (Product n)) ^2) + ((Im (Product n)) ^2)) is V31() real ext-real Element of REAL
S is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
len S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Product S is V31() real ext-real Element of REAL
<*> the carrier of F_Complex is Relation-like non-empty empty-yielding NAT -defined the carrier of F_Complex -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of the carrier of F_Complex *
the carrier of F_Complex * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
n is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom n is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Product n is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) associative Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
K228( the carrier of F_Complex,n, the multF of F_Complex) is V31() right_complementable Element of the carrier of F_Complex
|.(Product n).| is V31() real ext-real Element of REAL
Re (Product n) is V31() real ext-real Element of REAL
(Re (Product n)) ^2 is V31() real ext-real Element of REAL
K104((Re (Product n)),(Re (Product n))) is V31() real ext-real set
Im (Product n) is V31() real ext-real Element of REAL
(Im (Product n)) ^2 is V31() real ext-real Element of REAL
K104((Im (Product n)),(Im (Product n))) is V31() real ext-real set
((Re (Product n)) ^2) + ((Im (Product n)) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (Product n)) ^2) + ((Im (Product n)) ^2)) is V31() real ext-real Element of REAL
S is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
len S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Product S is V31() real ext-real Element of REAL
bool the carrier of F_Complex is set
cMGFC is non empty finite Element of bool the carrier of F_Complex
card cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
canFS cMGFC is Relation-like NAT -defined cMGFC -valued Function-like one-to-one non empty V26(cMGFC) finite FinSequence-like FinSubsequence-like finite-support FinSequence of cMGFC
(cMGFC,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
poly_with_roots ((cMGFC,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
[:NAT, the carrier of F_Complex:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of F_Complex:] is non empty non trivial non finite set
n is V31() right_complementable Element of the carrier of F_Complex
eval ((poly_with_roots ((cMGFC,1) -bag)),n) is V31() right_complementable Element of the carrier of F_Complex
|.(eval ((poly_with_roots ((cMGFC,1) -bag)),n)).| is V31() real ext-real Element of REAL
Re (eval ((poly_with_roots ((cMGFC,1) -bag)),n)) is V31() real ext-real Element of REAL
(Re (eval ((poly_with_roots ((cMGFC,1) -bag)),n))) ^2 is V31() real ext-real Element of REAL
K104((Re (eval ((poly_with_roots ((cMGFC,1) -bag)),n))),(Re (eval ((poly_with_roots ((cMGFC,1) -bag)),n)))) is V31() real ext-real set
Im (eval ((poly_with_roots ((cMGFC,1) -bag)),n)) is V31() real ext-real Element of REAL
(Im (eval ((poly_with_roots ((cMGFC,1) -bag)),n))) ^2 is V31() real ext-real Element of REAL
K104((Im (eval ((poly_with_roots ((cMGFC,1) -bag)),n))),(Im (eval ((poly_with_roots ((cMGFC,1) -bag)),n)))) is V31() real ext-real set
((Re (eval ((poly_with_roots ((cMGFC,1) -bag)),n))) ^2) + ((Im (eval ((poly_with_roots ((cMGFC,1) -bag)),n))) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (eval ((poly_with_roots ((cMGFC,1) -bag)),n))) ^2) + ((Im (eval ((poly_with_roots ((cMGFC,1) -bag)),n))) ^2)) is V31() real ext-real Element of REAL
S is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
len S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom S is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Product S is V31() real ext-real Element of REAL
len (canFS cMGFC) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom (canFS cMGFC) is non empty finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Seg (card cMGFC) is non empty finite card cMGFC -element V67() V68() V69() V70() V71() V72() Element of bool NAT
fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(canFS cMGFC) . fs is set
qc is V31() right_complementable Element of the carrier of F_Complex
- qc is V31() right_complementable Element of the carrier of F_Complex
<%(- qc),(1_ F_Complex)%> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (<%(- qc),(1_ F_Complex)%>,n) is V31() right_complementable Element of the carrier of F_Complex
p1 is V31() right_complementable Element of the carrier of F_Complex
fs is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qc is V31() right_complementable Element of the carrier of F_Complex
(canFS cMGFC) . q is set
fs /. q is V31() right_complementable Element of the carrier of F_Complex
fs . q is set
- qc is V31() right_complementable Element of the carrier of F_Complex
<%(- qc),(1_ F_Complex)%> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (<%(- qc),(1_ F_Complex)%>,n) is V31() right_complementable Element of the carrier of F_Complex
p1 is V31() right_complementable Element of the carrier of F_Complex
- p1 is V31() right_complementable Element of the carrier of F_Complex
<%(- p1),(1_ F_Complex)%> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (<%(- p1),(1_ F_Complex)%>,n) is V31() right_complementable Element of the carrier of F_Complex
q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs /. q is V31() right_complementable Element of the carrier of F_Complex
|.(fs /. q).| is V31() real ext-real Element of REAL
Re (fs /. q) is V31() real ext-real Element of REAL
(Re (fs /. q)) ^2 is V31() real ext-real Element of REAL
K104((Re (fs /. q)),(Re (fs /. q))) is V31() real ext-real set
Im (fs /. q) is V31() real ext-real Element of REAL
(Im (fs /. q)) ^2 is V31() real ext-real Element of REAL
K104((Im (fs /. q)),(Im (fs /. q))) is V31() real ext-real set
((Re (fs /. q)) ^2) + ((Im (fs /. q)) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (fs /. q)) ^2) + ((Im (fs /. q)) ^2)) is V31() real ext-real Element of REAL
S . q is V31() real ext-real set
(canFS cMGFC) . q is set
fs . q is set
p1 is V31() right_complementable Element of the carrier of F_Complex
- p1 is V31() right_complementable Element of the carrier of F_Complex
<%(- p1),(1_ F_Complex)%> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (<%(- p1),(1_ F_Complex)%>,n) is V31() right_complementable Element of the carrier of F_Complex
(- p1) + n is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
the addF of F_Complex . ((- p1),n) is V31() right_complementable Element of the carrier of F_Complex
[(- p1),n] is non empty set
{(- p1),n} is non empty finite V67() set
{(- p1)} is non empty trivial finite 1 -element V67() set
{{(- p1),n},{(- p1)}} is non empty finite V44() set
the addF of F_Complex . [(- p1),n] is set
(- p1) + n is V31() Element of COMPLEX
n - p1 is V31() right_complementable Element of the carrier of F_Complex
n + (- p1) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex . (n,(- p1)) is V31() right_complementable Element of the carrier of F_Complex
[n,(- p1)] is non empty set
{n,(- p1)} is non empty finite V67() set
{n} is non empty trivial finite 1 -element V67() set
{{n,(- p1)},{n}} is non empty finite V44() set
the addF of F_Complex . [n,(- p1)] is set
n + (- p1) is V31() Element of COMPLEX
len fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Product fs is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) associative Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
K228( the carrier of F_Complex,fs, the multF of F_Complex) is V31() right_complementable Element of the carrier of F_Complex
cMGFC is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
dom cMGFC is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Sum cMGFC is V31() right_complementable Element of the carrier of F_Complex
n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom S is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Sum S is V31() right_complementable Element of the carrier of F_Complex
fs is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
q is V31() right_complementable Element of the carrier of F_Complex
<*q*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex *
the carrier of F_Complex * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
fs ^ <*q*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S . qc is set
fs . qc is set
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S . (len S) is set
Sum fs is V31() right_complementable Element of the carrier of F_Complex
len fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
len <*q*> is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len fs) + (len <*q*>) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len fs) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qc is V31() Element of COMPLEX
p1 is V31() Element of COMPLEX
Sum <*q*> is V31() right_complementable Element of the carrier of F_Complex
(Sum fs) + (Sum <*q*>) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
the addF of F_Complex . ((Sum fs),(Sum <*q*>)) is V31() right_complementable Element of the carrier of F_Complex
[(Sum fs),(Sum <*q*>)] is non empty set
{(Sum fs),(Sum <*q*>)} is non empty finite V67() set
{(Sum fs)} is non empty trivial finite 1 -element V67() set
{{(Sum fs),(Sum <*q*>)},{(Sum fs)}} is non empty finite V44() set
the addF of F_Complex . [(Sum fs),(Sum <*q*>)] is set
(Sum fs) + (Sum <*q*>) is V31() Element of COMPLEX
ps is V31() real ext-real integer set
qi is V31() real ext-real integer set
ps + qi is V31() real ext-real integer rational Element of INT
len cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom n is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Sum n is V31() right_complementable Element of the carrier of F_Complex
<*> the carrier of F_Complex is Relation-like non-empty empty-yielding NAT -defined the carrier of F_Complex -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of the carrier of F_Complex *
the carrier of F_Complex * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
n is V31() real ext-real Element of REAL
MGFC is V31() right_complementable Element of the carrier of F_Complex
S is V31() real ext-real Element of REAL
cMGFC is V31() right_complementable Element of the carrier of F_Complex
n * S is V31() real ext-real Element of REAL
MGFC * cMGFC is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) associative Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
the multF of F_Complex . (MGFC,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
[MGFC,cMGFC] is non empty set
{MGFC,cMGFC} is non empty finite V67() set
{MGFC} is non empty trivial finite 1 -element V67() set
{{MGFC,cMGFC},{MGFC}} is non empty finite V44() set
the multF of F_Complex . [MGFC,cMGFC] is set
MGFC * cMGFC is V31() Element of COMPLEX
n + S is V31() real ext-real Element of REAL
MGFC + cMGFC is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . (MGFC,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex . [MGFC,cMGFC] is set
MGFC + cMGFC is V31() Element of COMPLEX
[**1,0**] is V31() right_complementable Element of the carrier of F_Complex
K104(0,<i>) is V31() set
K103(1,K104(0,<i>)) is V31() set
MGFC is V31() real ext-real Element of REAL
[**MGFC,0**] is V31() right_complementable Element of the carrier of F_Complex
K103(MGFC,K104(0,<i>)) is V31() set
MGFC - 1 is V31() real ext-real Element of REAL
cMGFC is V31() right_complementable Element of the carrier of F_Complex
|.cMGFC.| is V31() real ext-real Element of REAL
Re cMGFC is V31() real ext-real Element of REAL
(Re cMGFC) ^2 is V31() real ext-real Element of REAL
K104((Re cMGFC),(Re cMGFC)) is V31() real ext-real set
Im cMGFC is V31() real ext-real Element of REAL
(Im cMGFC) ^2 is V31() real ext-real Element of REAL
K104((Im cMGFC),(Im cMGFC)) is V31() real ext-real set
((Re cMGFC) ^2) + ((Im cMGFC) ^2) is V31() real ext-real Element of REAL
sqrt (((Re cMGFC) ^2) + ((Im cMGFC) ^2)) is V31() real ext-real Element of REAL
[**MGFC,0**] - cMGFC is V31() right_complementable Element of the carrier of F_Complex
- cMGFC is V31() right_complementable Element of the carrier of F_Complex
[**MGFC,0**] + (- cMGFC) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
the addF of F_Complex . ([**MGFC,0**],(- cMGFC)) is V31() right_complementable Element of the carrier of F_Complex
[[**MGFC,0**],(- cMGFC)] is non empty set
{[**MGFC,0**],(- cMGFC)} is non empty finite V67() set
{[**MGFC,0**]} is non empty trivial finite 1 -element V67() set
{{[**MGFC,0**],(- cMGFC)},{[**MGFC,0**]}} is non empty finite V44() set
the addF of F_Complex . [[**MGFC,0**],(- cMGFC)] is set
[**MGFC,0**] + (- cMGFC) is V31() Element of COMPLEX
|.([**MGFC,0**] - cMGFC).| is V31() real ext-real Element of REAL
Re ([**MGFC,0**] - cMGFC) is V31() real ext-real Element of REAL
(Re ([**MGFC,0**] - cMGFC)) ^2 is V31() real ext-real Element of REAL
K104((Re ([**MGFC,0**] - cMGFC)),(Re ([**MGFC,0**] - cMGFC))) is V31() real ext-real set
Im ([**MGFC,0**] - cMGFC) is V31() real ext-real Element of REAL
(Im ([**MGFC,0**] - cMGFC)) ^2 is V31() real ext-real Element of REAL
K104((Im ([**MGFC,0**] - cMGFC)),(Im ([**MGFC,0**] - cMGFC))) is V31() real ext-real set
((Re ([**MGFC,0**] - cMGFC)) ^2) + ((Im ([**MGFC,0**] - cMGFC)) ^2) is V31() real ext-real Element of REAL
sqrt (((Re ([**MGFC,0**] - cMGFC)) ^2) + ((Im ([**MGFC,0**] - cMGFC)) ^2)) is V31() real ext-real Element of REAL
((Re cMGFC) ^2) + ((Im cMGFC) ^2) is V31() real ext-real Element of REAL
1 ^2 is V31() real ext-real Element of REAL
K104(1,1) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal set
2 * MGFC is V31() real ext-real Element of REAL
(2 * MGFC) * (Re cMGFC) is V31() real ext-real Element of REAL
- ((2 * MGFC) * (Re cMGFC)) is V31() real ext-real Element of REAL
- (2 * MGFC) is V31() real ext-real Element of REAL
MGFC ^2 is V31() real ext-real Element of REAL
K104(MGFC,MGFC) is V31() real ext-real set
(- ((2 * MGFC) * (Re cMGFC))) + (MGFC ^2) is V31() real ext-real Element of REAL
(- (2 * MGFC)) + (MGFC ^2) is V31() real ext-real Element of REAL
(MGFC ^2) - ((2 * MGFC) * (Re cMGFC)) is V31() real ext-real Element of REAL
((MGFC ^2) - ((2 * MGFC) * (Re cMGFC))) + 1 is V31() real ext-real Element of REAL
(MGFC ^2) - (2 * MGFC) is V31() real ext-real Element of REAL
((MGFC ^2) - (2 * MGFC)) + 1 is V31() real ext-real Element of REAL
MGFC - (Re cMGFC) is V31() real ext-real Element of REAL
- (Im cMGFC) is V31() real ext-real Element of REAL
[**(MGFC - (Re cMGFC)),(- (Im cMGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((- (Im cMGFC)),<i>) is V31() set
K103((MGFC - (Re cMGFC)),K104((- (Im cMGFC)),<i>)) is V31() set
Re [**(MGFC - (Re cMGFC)),(- (Im cMGFC))**] is V31() real ext-real Element of REAL
Im [**(MGFC - (Re cMGFC)),(- (Im cMGFC))**] is V31() real ext-real Element of REAL
fs is V31() right_complementable Element of the carrier of F_Complex
fs - cMGFC is V31() right_complementable Element of the carrier of F_Complex
fs + (- cMGFC) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex . (fs,(- cMGFC)) is V31() right_complementable Element of the carrier of F_Complex
[fs,(- cMGFC)] is non empty set
{fs,(- cMGFC)} is non empty finite V67() set
{fs} is non empty trivial finite 1 -element V67() set
{{fs,(- cMGFC)},{fs}} is non empty finite V44() set
the addF of F_Complex . [fs,(- cMGFC)] is set
fs + (- cMGFC) is V31() Element of COMPLEX
|.(fs - cMGFC).| is V31() real ext-real Element of REAL
Re (fs - cMGFC) is V31() real ext-real Element of REAL
(Re (fs - cMGFC)) ^2 is V31() real ext-real Element of REAL
K104((Re (fs - cMGFC)),(Re (fs - cMGFC))) is V31() real ext-real set
Im (fs - cMGFC) is V31() real ext-real Element of REAL
(Im (fs - cMGFC)) ^2 is V31() real ext-real Element of REAL
K104((Im (fs - cMGFC)),(Im (fs - cMGFC))) is V31() real ext-real set
((Re (fs - cMGFC)) ^2) + ((Im (fs - cMGFC)) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (fs - cMGFC)) ^2) + ((Im (fs - cMGFC)) ^2)) is V31() real ext-real Element of REAL
|.(fs - cMGFC).| ^2 is V31() real ext-real Element of REAL
K104(|.(fs - cMGFC).|,|.(fs - cMGFC).|) is V31() real ext-real set
[**(Re cMGFC),(Im cMGFC)**] is V31() right_complementable Element of the carrier of F_Complex
K104((Im cMGFC),<i>) is V31() set
K103((Re cMGFC),K104((Im cMGFC),<i>)) is V31() set
[**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**] is V31() right_complementable Element of the carrier of F_Complex
- [**(Re cMGFC),(Im cMGFC)**] is V31() right_complementable Element of the carrier of F_Complex
[**MGFC,0**] + (- [**(Re cMGFC),(Im cMGFC)**]) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex . ([**MGFC,0**],(- [**(Re cMGFC),(Im cMGFC)**])) is V31() right_complementable Element of the carrier of F_Complex
[[**MGFC,0**],(- [**(Re cMGFC),(Im cMGFC)**])] is non empty set
{[**MGFC,0**],(- [**(Re cMGFC),(Im cMGFC)**])} is non empty finite V67() set
{{[**MGFC,0**],(- [**(Re cMGFC),(Im cMGFC)**])},{[**MGFC,0**]}} is non empty finite V44() set
the addF of F_Complex . [[**MGFC,0**],(- [**(Re cMGFC),(Im cMGFC)**])] is set
[**MGFC,0**] + (- [**(Re cMGFC),(Im cMGFC)**]) is V31() Element of COMPLEX
|.([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**]).| is V31() real ext-real Element of REAL
Re ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**]) is V31() real ext-real Element of REAL
(Re ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**])) ^2 is V31() real ext-real Element of REAL
K104((Re ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**])),(Re ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**]))) is V31() real ext-real set
Im ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**]) is V31() real ext-real Element of REAL
(Im ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**])) ^2 is V31() real ext-real Element of REAL
K104((Im ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**])),(Im ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**]))) is V31() real ext-real set
((Re ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**])) ^2) + ((Im ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**])) ^2) is V31() real ext-real Element of REAL
sqrt (((Re ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**])) ^2) + ((Im ([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**])) ^2)) is V31() real ext-real Element of REAL
|.([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**]).| ^2 is V31() real ext-real Element of REAL
K104(|.([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**]).|,|.([**MGFC,0**] - [**(Re cMGFC),(Im cMGFC)**]).|) is V31() real ext-real set
0 - (Im cMGFC) is V31() real ext-real Element of REAL
[**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((0 - (Im cMGFC)),<i>) is V31() set
K103((MGFC - (Re cMGFC)),K104((0 - (Im cMGFC)),<i>)) is V31() set
|.[**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**].| is V31() real ext-real Element of REAL
Re [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**] is V31() real ext-real Element of REAL
(Re [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**]) ^2 is V31() real ext-real Element of REAL
K104((Re [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**]),(Re [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**])) is V31() real ext-real set
Im [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**] is V31() real ext-real Element of REAL
(Im [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**]) ^2 is V31() real ext-real Element of REAL
K104((Im [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**]),(Im [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**])) is V31() real ext-real set
((Re [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**]) ^2) + ((Im [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**]) ^2) is V31() real ext-real Element of REAL
sqrt (((Re [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**]) ^2) + ((Im [**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**]) ^2)) is V31() real ext-real Element of REAL
|.[**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**].| ^2 is V31() real ext-real Element of REAL
K104(|.[**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**].|,|.[**(MGFC - (Re cMGFC)),(0 - (Im cMGFC))**].|) is V31() real ext-real set
(MGFC - (Re cMGFC)) ^2 is V31() real ext-real Element of REAL
K104((MGFC - (Re cMGFC)),(MGFC - (Re cMGFC))) is V31() real ext-real set
((MGFC - (Re cMGFC)) ^2) + ((Im cMGFC) ^2) is V31() real ext-real Element of REAL
(MGFC - 1) ^2 is V31() real ext-real Element of REAL
K104((MGFC - 1),(MGFC - 1)) is V31() real ext-real set
MGFC is Relation-like NAT -defined REAL -valued Function-like non empty finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
dom MGFC is non empty finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Product MGFC is V31() real ext-real Element of REAL
cMGFC is V31() real ext-real Element of REAL
n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
S is set
<*S*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
n ^ <*S*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like finite-support set
rng <*S*> is non empty trivial finite 1 -element set
{S} is non empty trivial finite 1 -element set
fs is V31() real ext-real Element of REAL
q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qc is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
len qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom qc is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Product qc is V31() real ext-real Element of REAL
(Product qc) * fs is V31() real ext-real Element of REAL
p1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
ps is set
<*ps*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
p1 ^ <*ps*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like finite-support set
i is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
rng i is finite V67() V68() V69() Element of bool REAL
{ps} is non empty trivial finite 1 -element set
qi is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
len qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
len i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len qi) + (len i) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len qi) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qc . (len qc) is V31() real ext-real set
x is V31() real ext-real Element of REAL
Seg (len qc) is finite len qc -element V67() V68() V69() V70() V71() V72() Element of bool NAT
cMGFC * x is V31() real ext-real Element of REAL
dom qi is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
lc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qi . lc is V31() real ext-real set
qc . lc is V31() real ext-real set
Product qi is V31() real ext-real Element of REAL
(Product qi) * x is V31() real ext-real Element of REAL
(Product qi) * fs is V31() real ext-real Element of REAL
((Product qi) * fs) * x is V31() real ext-real Element of REAL
len MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg (len MGFC) is non empty finite len MGFC -element V67() V68() V69() V70() V71() V72() Element of bool NAT
q is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
dom q is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q . qc is V31() real ext-real set
MGFC . qc is V31() real ext-real set
len q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
len n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
len <*S*> is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len n) + (len <*S*>) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len n) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC . (len MGFC) is V31() real ext-real set
qc is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
len qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom qc is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Product qc is V31() real ext-real Element of REAL
(Product qc) * fs is V31() real ext-real Element of REAL
<*> REAL is Relation-like non-empty empty-yielding NAT -defined REAL -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of REAL *
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
Product q is V31() real ext-real Element of REAL
(Product q) * fs is V31() real ext-real Element of REAL
MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power F_Complex) . ((1_ F_Complex),MGFC) is V31() right_complementable Element of the carrier of F_Complex
[(1_ F_Complex),MGFC] is non empty set
{(1_ F_Complex),MGFC} is non empty finite V67() set
{(1_ F_Complex)} is non empty trivial finite 1 -element V67() set
{{(1_ F_Complex),MGFC},{(1_ F_Complex)}} is non empty finite V44() set
(power F_Complex) . [(1_ F_Complex),MGFC] is set
1 to_power MGFC is V31() real ext-real Element of REAL
1 |^ MGFC is V31() real ext-real integer set
[**(1 to_power MGFC),0**] is V31() right_complementable Element of the carrier of F_Complex
K103((1 to_power MGFC),K104(0,<i>)) is V31() set
2 * PI is V31() real ext-real Element of REAL
cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * cMGFC is V31() real ext-real Element of REAL
MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
((2 * PI) * cMGFC) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * cMGFC) / MGFC) is V31() real ext-real Element of REAL
cMGFC mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * (cMGFC mod MGFC) is V31() real ext-real Element of REAL
((2 * PI) * (cMGFC mod MGFC)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * cMGFC) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
cMGFC div MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC * (cMGFC div MGFC) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC * (cMGFC div MGFC)) + (cMGFC mod MGFC) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * (MGFC * (cMGFC div MGFC)) is V31() real ext-real Element of REAL
((2 * PI) * (MGFC * (cMGFC div MGFC))) + ((2 * PI) * (cMGFC mod MGFC)) is V31() real ext-real Element of REAL
(((2 * PI) * (MGFC * (cMGFC div MGFC))) + ((2 * PI) * (cMGFC mod MGFC))) / MGFC is V31() real ext-real Element of REAL
MGFC * 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
((2 * PI) * (MGFC * (cMGFC div MGFC))) / (MGFC * 1) is V31() real ext-real Element of REAL
(((2 * PI) * (MGFC * (cMGFC div MGFC))) / (MGFC * 1)) + (((2 * PI) * (cMGFC mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
(2 * PI) / MGFC is V31() real ext-real Element of REAL
((2 * PI) / MGFC) * (MGFC * (cMGFC div MGFC)) is V31() real ext-real Element of REAL
(((2 * PI) / MGFC) * (MGFC * (cMGFC div MGFC))) / 1 is V31() real ext-real Element of REAL
((((2 * PI) / MGFC) * (MGFC * (cMGFC div MGFC))) / 1) + (((2 * PI) * (cMGFC mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
1 / MGFC is V31() real ext-real non negative rational Element of RAT
(2 * PI) * (1 / MGFC) is V31() real ext-real Element of REAL
((2 * PI) * (1 / MGFC)) * (MGFC * (cMGFC div MGFC)) is V31() real ext-real Element of REAL
(((2 * PI) * (1 / MGFC)) * (MGFC * (cMGFC div MGFC))) + (((2 * PI) * (cMGFC mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
(cMGFC div MGFC) * MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(1 / MGFC) * ((cMGFC div MGFC) * MGFC) is V31() real ext-real non negative rational Element of RAT
(2 * PI) * ((1 / MGFC) * ((cMGFC div MGFC) * MGFC)) is V31() real ext-real Element of REAL
((2 * PI) * ((1 / MGFC) * ((cMGFC div MGFC) * MGFC))) + (((2 * PI) * (cMGFC mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
(cMGFC div MGFC) * 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * ((cMGFC div MGFC) * 1) is V31() real ext-real Element of REAL
((2 * PI) * ((cMGFC div MGFC) * 1)) + (((2 * PI) * (cMGFC mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
(2 * PI) * (cMGFC div MGFC) is V31() real ext-real Element of REAL
((2 * PI) * (cMGFC div MGFC)) + (((2 * PI) * (cMGFC mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
((2 * PI) * (cMGFC div MGFC)) + 0 is V31() real ext-real Element of REAL
sin (((2 * PI) * (cMGFC div MGFC)) + 0) is V31() real ext-real Element of REAL
(sin (((2 * PI) * (cMGFC div MGFC)) + 0)) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
cos (((2 * PI) * (cMGFC div MGFC)) + 0) is V31() real ext-real Element of REAL
(cos (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
((sin (((2 * PI) * (cMGFC div MGFC)) + 0)) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) + ((cos (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) is V31() real ext-real Element of REAL
sin . (((2 * PI) * (cMGFC div MGFC)) + 0) is V31() real ext-real Element of REAL
(sin . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
((sin . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) + ((cos (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) is V31() real ext-real Element of REAL
cos . (((2 * PI) * (cMGFC div MGFC)) + 0) is V31() real ext-real Element of REAL
(cos . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
((sin . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) + ((cos . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) is V31() real ext-real Element of REAL
sin . 0 is V31() real ext-real Element of REAL
(sin . 0) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
((sin . 0) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) + ((cos . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) is V31() real ext-real Element of REAL
cos . 0 is V31() real ext-real Element of REAL
(cos . 0) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
((sin . 0) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) + ((cos . 0) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) is V31() real ext-real Element of REAL
(cos (((2 * PI) * (cMGFC div MGFC)) + 0)) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
(sin (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
((cos (((2 * PI) * (cMGFC div MGFC)) + 0)) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) - ((sin (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) is V31() real ext-real Element of REAL
(cos . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
((cos . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) - ((sin (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) is V31() real ext-real Element of REAL
(sin . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
((cos . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) - ((sin . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) is V31() real ext-real Element of REAL
(cos . 0) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
((cos . 0) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) - ((sin . (((2 * PI) * (cMGFC div MGFC)) + 0)) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) is V31() real ext-real Element of REAL
(sin . 0) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC)) is V31() real ext-real Element of REAL
((cos . 0) * (cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) - ((sin . 0) * (sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))) is V31() real ext-real Element of REAL
cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * cMGFC is V31() real ext-real Element of REAL
MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
((2 * PI) * cMGFC) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * cMGFC) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * cMGFC) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * cMGFC) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * cMGFC) / MGFC)),K104((sin (((2 * PI) * cMGFC) / MGFC)),<i>)) is V31() set
cMGFC mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * (cMGFC mod MGFC) is V31() real ext-real Element of REAL
((2 * PI) * (cMGFC mod MGFC)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC)),(sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC)),K104((sin (((2 * PI) * (cMGFC mod MGFC)) / MGFC)),<i>)) is V31() set
[**(cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K103((cos (((2 * PI) * (cMGFC mod MGFC)) / MGFC)),K104((sin (((2 * PI) * cMGFC) / MGFC)),<i>)) is V31() set
cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * cMGFC is V31() real ext-real Element of REAL
MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
((2 * PI) * cMGFC) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * cMGFC) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * cMGFC) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * cMGFC) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * cMGFC) / MGFC)),K104((sin (((2 * PI) * cMGFC) / MGFC)),<i>)) is V31() set
n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * n is V31() real ext-real Element of REAL
((2 * PI) * n) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * n) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * n) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * n) / MGFC)),(sin (((2 * PI) * n) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * n) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * n) / MGFC)),K104((sin (((2 * PI) * n) / MGFC)),<i>)) is V31() set
[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**] * [**(cos (((2 * PI) * n) / MGFC)),(sin (((2 * PI) * n) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) associative Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
the multF of F_Complex . ([**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**],[**(cos (((2 * PI) * n) / MGFC)),(sin (((2 * PI) * n) / MGFC))**]) is V31() right_complementable Element of the carrier of F_Complex
[[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**],[**(cos (((2 * PI) * n) / MGFC)),(sin (((2 * PI) * n) / MGFC))**]] is non empty set
{[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**],[**(cos (((2 * PI) * n) / MGFC)),(sin (((2 * PI) * n) / MGFC))**]} is non empty finite V67() set
{[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**]} is non empty trivial finite 1 -element V67() set
{{[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**],[**(cos (((2 * PI) * n) / MGFC)),(sin (((2 * PI) * n) / MGFC))**]},{[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**]}} is non empty finite V44() set
the multF of F_Complex . [[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**],[**(cos (((2 * PI) * n) / MGFC)),(sin (((2 * PI) * n) / MGFC))**]] is set
[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**] * [**(cos (((2 * PI) * n) / MGFC)),(sin (((2 * PI) * n) / MGFC))**] is V31() Element of COMPLEX
cMGFC + n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC + n) mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * ((cMGFC + n) mod MGFC) is V31() real ext-real Element of REAL
((2 * PI) * ((cMGFC + n) mod MGFC)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * ((cMGFC + n) mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * ((cMGFC + n) mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * ((cMGFC + n) mod MGFC)) / MGFC)),(sin (((2 * PI) * ((cMGFC + n) mod MGFC)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * ((cMGFC + n) mod MGFC)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * ((cMGFC + n) mod MGFC)) / MGFC)),K104((sin (((2 * PI) * ((cMGFC + n) mod MGFC)) / MGFC)),<i>)) is V31() set
(((2 * PI) * cMGFC) / MGFC) + (((2 * PI) * n) / MGFC) is V31() real ext-real Element of REAL
((2 * PI) * cMGFC) + ((2 * PI) * n) is V31() real ext-real Element of REAL
(((2 * PI) * cMGFC) + ((2 * PI) * n)) / MGFC is V31() real ext-real Element of REAL
(2 * PI) * (cMGFC + n) is V31() real ext-real Element of REAL
((2 * PI) * (cMGFC + n)) / MGFC is V31() real ext-real Element of REAL
(cos (((2 * PI) * cMGFC) / MGFC)) * (cos (((2 * PI) * n) / MGFC)) is V31() real ext-real Element of REAL
(sin (((2 * PI) * cMGFC) / MGFC)) * (sin (((2 * PI) * n) / MGFC)) is V31() real ext-real Element of REAL
((cos (((2 * PI) * cMGFC) / MGFC)) * (cos (((2 * PI) * n) / MGFC))) - ((sin (((2 * PI) * cMGFC) / MGFC)) * (sin (((2 * PI) * n) / MGFC))) is V31() real ext-real Element of REAL
cos (((2 * PI) * (cMGFC + n)) / MGFC) is V31() real ext-real Element of REAL
(cos (((2 * PI) * cMGFC) / MGFC)) * (sin (((2 * PI) * n) / MGFC)) is V31() real ext-real Element of REAL
(cos (((2 * PI) * n) / MGFC)) * (sin (((2 * PI) * cMGFC) / MGFC)) is V31() real ext-real Element of REAL
((cos (((2 * PI) * cMGFC) / MGFC)) * (sin (((2 * PI) * n) / MGFC))) + ((cos (((2 * PI) * n) / MGFC)) * (sin (((2 * PI) * cMGFC) / MGFC))) is V31() real ext-real Element of REAL
sin (((2 * PI) * (cMGFC + n)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * (cMGFC + n)) / MGFC)),(sin (((2 * PI) * (cMGFC + n)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * (cMGFC + n)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * (cMGFC + n)) / MGFC)),K104((sin (((2 * PI) * (cMGFC + n)) / MGFC)),<i>)) is V31() set
MGFC is non empty unital associative multMagma
the carrier of MGFC is non empty set
power MGFC is Relation-like [: the carrier of MGFC,NAT:] -defined the carrier of MGFC -valued Function-like V25([: the carrier of MGFC,NAT:], the carrier of MGFC) Element of bool [:[: the carrier of MGFC,NAT:], the carrier of MGFC:]
[: the carrier of MGFC,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() V60() set
[:[: the carrier of MGFC,NAT:], the carrier of MGFC:] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of MGFC,NAT:], the carrier of MGFC:] is non empty non trivial non finite set
cMGFC is Element of the carrier of MGFC
n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power MGFC) . (cMGFC,n) is Element of the carrier of MGFC
[cMGFC,n] is non empty set
{cMGFC,n} is non empty finite set
{cMGFC} is non empty trivial finite 1 -element set
{{cMGFC,n},{cMGFC}} is non empty finite V44() set
(power MGFC) . [cMGFC,n] is set
fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n * fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power MGFC) . (cMGFC,(n * fs)) is Element of the carrier of MGFC
[cMGFC,(n * fs)] is non empty set
{cMGFC,(n * fs)} is non empty finite set
{{cMGFC,(n * fs)},{cMGFC}} is non empty finite V44() set
(power MGFC) . [cMGFC,(n * fs)] is set
(power MGFC) . (((power MGFC) . (cMGFC,n)),fs) is Element of the carrier of MGFC
[((power MGFC) . (cMGFC,n)),fs] is non empty set
{((power MGFC) . (cMGFC,n)),fs} is non empty finite set
{((power MGFC) . (cMGFC,n))} is non empty trivial finite 1 -element set
{{((power MGFC) . (cMGFC,n)),fs},{((power MGFC) . (cMGFC,n))}} is non empty finite V44() set
(power MGFC) . [((power MGFC) . (cMGFC,n)),fs] is set
fs + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n * (fs + 1) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power MGFC) . (cMGFC,(n * (fs + 1))) is Element of the carrier of MGFC
[cMGFC,(n * (fs + 1))] is non empty set
{cMGFC,(n * (fs + 1))} is non empty finite set
{{cMGFC,(n * (fs + 1))},{cMGFC}} is non empty finite V44() set
(power MGFC) . [cMGFC,(n * (fs + 1))] is set
(power MGFC) . (((power MGFC) . (cMGFC,n)),(fs + 1)) is Element of the carrier of MGFC
[((power MGFC) . (cMGFC,n)),(fs + 1)] is non empty set
{((power MGFC) . (cMGFC,n)),(fs + 1)} is non empty finite set
{{((power MGFC) . (cMGFC,n)),(fs + 1)},{((power MGFC) . (cMGFC,n))}} is non empty finite V44() set
(power MGFC) . [((power MGFC) . (cMGFC,n)),(fs + 1)] is set
n * 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n * fs) + (n * 1) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power MGFC) . (cMGFC,((n * fs) + (n * 1))) is Element of the carrier of MGFC
[cMGFC,((n * fs) + (n * 1))] is non empty set
{cMGFC,((n * fs) + (n * 1))} is non empty finite set
{{cMGFC,((n * fs) + (n * 1))},{cMGFC}} is non empty finite V44() set
(power MGFC) . [cMGFC,((n * fs) + (n * 1))] is set
((power MGFC) . (cMGFC,(n * fs))) * ((power MGFC) . (cMGFC,n)) is Element of the carrier of MGFC
the multF of MGFC is Relation-like [: the carrier of MGFC, the carrier of MGFC:] -defined the carrier of MGFC -valued Function-like V25([: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC) associative Element of bool [:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:]
[: the carrier of MGFC, the carrier of MGFC:] is Relation-like set
[:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:] is Relation-like set
bool [:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:] is set
the multF of MGFC . (((power MGFC) . (cMGFC,(n * fs))),((power MGFC) . (cMGFC,n))) is Element of the carrier of MGFC
[((power MGFC) . (cMGFC,(n * fs))),((power MGFC) . (cMGFC,n))] is non empty set
{((power MGFC) . (cMGFC,(n * fs))),((power MGFC) . (cMGFC,n))} is non empty finite set
{((power MGFC) . (cMGFC,(n * fs)))} is non empty trivial finite 1 -element set
{{((power MGFC) . (cMGFC,(n * fs))),((power MGFC) . (cMGFC,n))},{((power MGFC) . (cMGFC,(n * fs)))}} is non empty finite V44() set
the multF of MGFC . [((power MGFC) . (cMGFC,(n * fs))),((power MGFC) . (cMGFC,n))] is set
n * 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered rational V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of NAT
(power MGFC) . (cMGFC,(n * 0)) is Element of the carrier of MGFC
[cMGFC,(n * 0)] is non empty set
{cMGFC,(n * 0)} is non empty finite set
{{cMGFC,(n * 0)},{cMGFC}} is non empty finite V44() set
(power MGFC) . [cMGFC,(n * 0)] is set
1_ MGFC is Element of the carrier of MGFC
(power MGFC) . (((power MGFC) . (cMGFC,n)),0) is Element of the carrier of MGFC
[((power MGFC) . (cMGFC,n)),0] is non empty set
{((power MGFC) . (cMGFC,n)),0} is non empty finite set
{((power MGFC) . (cMGFC,n))} is non empty trivial finite 1 -element set
{{((power MGFC) . (cMGFC,n)),0},{((power MGFC) . (cMGFC,n))}} is non empty finite V44() set
(power MGFC) . [((power MGFC) . (cMGFC,n)),0] is set
MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (cMGFC,MGFC) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,MGFC] is non empty set
{cMGFC,MGFC} is non empty finite V67() set
{cMGFC} is non empty trivial finite 1 -element V67() set
{{cMGFC,MGFC},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,MGFC] is set
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power F_Complex) . (cMGFC,S) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,S] is non empty set
{cMGFC,S} is non empty finite V67() set
{{cMGFC,S},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,S] is set
((power F_Complex) . (cMGFC,S)) * cMGFC is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) associative Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
the multF of F_Complex . (((power F_Complex) . (cMGFC,S)),cMGFC) is V31() right_complementable Element of the carrier of F_Complex
[((power F_Complex) . (cMGFC,S)),cMGFC] is non empty set
{((power F_Complex) . (cMGFC,S)),cMGFC} is non empty finite V67() set
{((power F_Complex) . (cMGFC,S))} is non empty trivial finite 1 -element V67() set
{{((power F_Complex) . (cMGFC,S)),cMGFC},{((power F_Complex) . (cMGFC,S))}} is non empty finite V44() set
the multF of F_Complex . [((power F_Complex) . (cMGFC,S)),cMGFC] is set
((power F_Complex) . (cMGFC,S)) * cMGFC is V31() Element of COMPLEX
n is V31() real ext-real integer set
fs is V31() real ext-real integer set
n * fs is V31() real ext-real integer rational Element of INT
S + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power F_Complex) . (cMGFC,(S + 1)) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,(S + 1)] is non empty set
{cMGFC,(S + 1)} is non empty finite V67() set
{{cMGFC,(S + 1)},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,(S + 1)] is set
(power F_Complex) . (cMGFC,0) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,0] is non empty set
{cMGFC,0} is non empty finite V67() set
{{cMGFC,0},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,0] is set
MGFC is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
dom MGFC is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Sum MGFC is V31() right_complementable Element of the carrier of F_Complex
cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom n is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Sum n is V31() right_complementable Element of the carrier of F_Complex
S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
fs is set
<*fs*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
S ^ <*fs*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like finite-support set
Seg (len n) is finite len n -element V67() V68() V69() V70() V71() V72() Element of bool NAT
q is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
rng q is finite set
{fs} is non empty trivial finite 1 -element set
qc is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
len q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len qc) + (len q) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len qc) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n . (len n) is set
p1 is V31() right_complementable Element of the carrier of F_Complex
dom qc is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qc . qi is set
n . qi is set
Sum qc is V31() right_complementable Element of the carrier of F_Complex
(Sum qc) + p1 is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
the addF of F_Complex . ((Sum qc),p1) is V31() right_complementable Element of the carrier of F_Complex
[(Sum qc),p1] is non empty set
{(Sum qc),p1} is non empty finite V67() set
{(Sum qc)} is non empty trivial finite 1 -element V67() set
{{(Sum qc),p1},{(Sum qc)}} is non empty finite V44() set
the addF of F_Complex . [(Sum qc),p1] is set
(Sum qc) + p1 is V31() Element of COMPLEX
qi is V31() real ext-real integer set
ps is V31() real ext-real integer set
qi + ps is V31() real ext-real integer rational Element of INT
cMGFC is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom cMGFC is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Sum cMGFC is V31() right_complementable Element of the carrier of F_Complex
0 -tuples_on the carrier of F_Complex is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
the carrier of F_Complex * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex * : len b₁ = 0 } is set
len MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() doubleLoopStr
NonZero MGFC is non empty Element of bool the carrier of MGFC
the carrier of MGFC is non empty non trivial set
bool the carrier of MGFC is set
[#] MGFC is non empty non proper Element of bool the carrier of MGFC
0. MGFC is zero right_complementable Element of the carrier of MGFC
the ZeroF of MGFC is right_complementable Element of the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element set
([#] MGFC) \ {(0. MGFC)} is Element of bool the carrier of MGFC
the multF of MGFC is Relation-like [: the carrier of MGFC, the carrier of MGFC:] -defined the carrier of MGFC -valued Function-like V25([: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC) associative Element of bool [:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:]
[: the carrier of MGFC, the carrier of MGFC:] is Relation-like set
[:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:] is Relation-like set
bool [:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:] is set
S is non empty set
the multF of MGFC || S is Relation-like Function-like set
[:S,S:] is Relation-like set
the multF of MGFC | [:S,S:] is Relation-like [:S,S:] -defined [: the carrier of MGFC, the carrier of MGFC:] -defined the carrier of MGFC -valued Function-like set
q is set
qc is set
p1 is set
[qc,p1] is non empty set
{qc,p1} is non empty finite set
{qc} is non empty trivial finite 1 -element set
{{qc,p1},{qc}} is non empty finite V44() set
[:[:S,S:], the carrier of MGFC:] is Relation-like set
bool [:[:S,S:], the carrier of MGFC:] is set
q is Relation-like [:S,S:] -defined the carrier of MGFC -valued Function-like V25([:S,S:], the carrier of MGFC) Element of bool [:[:S,S:], the carrier of MGFC:]
rng q is set
qc is set
dom q is Relation-like set
p1 is set
q . p1 is set
ps is set
qi is set
[ps,qi] is non empty set
{ps,qi} is non empty finite set
{ps} is non empty trivial finite 1 -element set
{{ps,qi},{ps}} is non empty finite V44() set
x is right_complementable Element of the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element Element of bool the carrier of MGFC
i is right_complementable Element of the carrier of MGFC
i * x is right_complementable Element of the carrier of MGFC
the multF of MGFC . (i,x) is right_complementable Element of the carrier of MGFC
[i,x] is non empty set
{i,x} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,x},{i}} is non empty finite V44() set
the multF of MGFC . [i,x] is set
[:[:S,S:],S:] is Relation-like set
bool [:[:S,S:],S:] is set
qc is Relation-like [:S,S:] -defined S -valued Function-like V25([:S,S:],S) Element of bool [:[:S,S:],S:]
multMagma(# S,qc #) is non empty strict multMagma
p1 is non empty multMagma
the carrier of p1 is non empty set
qi is right_complementable Element of the carrier of MGFC
x is Element of the carrier of p1
i is right_complementable Element of the carrier of MGFC
lc is Element of the carrier of p1
[x,lc] is non empty Element of [: the carrier of p1, the carrier of p1:]
[: the carrier of p1, the carrier of p1:] is Relation-like set
{x,lc} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,lc},{x}} is non empty finite V44() set
qi * i is right_complementable Element of the carrier of MGFC
the multF of MGFC . (qi,i) is right_complementable Element of the carrier of MGFC
[qi,i] is non empty set
{qi,i} is non empty finite set
{qi} is non empty trivial finite 1 -element set
{{qi,i},{qi}} is non empty finite V44() set
the multF of MGFC . [qi,i] is set
x * lc is Element of the carrier of p1
the multF of p1 is Relation-like [: the carrier of p1, the carrier of p1:] -defined the carrier of p1 -valued Function-like V25([: the carrier of p1, the carrier of p1:], the carrier of p1) Element of bool [:[: the carrier of p1, the carrier of p1:], the carrier of p1:]
[:[: the carrier of p1, the carrier of p1:], the carrier of p1:] is Relation-like set
bool [:[: the carrier of p1, the carrier of p1:], the carrier of p1:] is set
the multF of p1 . (x,lc) is Element of the carrier of p1
[x,lc] is non empty set
the multF of p1 . [x,lc] is set
1_ MGFC is right_complementable Element of the carrier of MGFC
1. MGFC is non zero right_complementable Element of the carrier of MGFC
the OneF of MGFC is right_complementable Element of the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element Element of bool the carrier of MGFC
qi is Element of the carrier of p1
i is Element of the carrier of p1
i * qi is Element of the carrier of p1
the multF of p1 . (i,qi) is Element of the carrier of p1
[i,qi] is non empty set
{i,qi} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,qi},{i}} is non empty finite V44() set
the multF of p1 . [i,qi] is set
qi * i is Element of the carrier of p1
the multF of p1 . (qi,i) is Element of the carrier of p1
[qi,i] is non empty set
{qi,i} is non empty finite set
{qi} is non empty trivial finite 1 -element set
{{qi,i},{qi}} is non empty finite V44() set
the multF of p1 . [qi,i] is set
x is right_complementable Element of the carrier of MGFC
x * (1_ MGFC) is right_complementable Element of the carrier of MGFC
the multF of MGFC . (x,(1_ MGFC)) is right_complementable Element of the carrier of MGFC
[x,(1_ MGFC)] is non empty set
{x,(1_ MGFC)} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,(1_ MGFC)},{x}} is non empty finite V44() set
the multF of MGFC . [x,(1_ MGFC)] is set
(1_ MGFC) * x is right_complementable Element of the carrier of MGFC
the multF of MGFC . ((1_ MGFC),x) is right_complementable Element of the carrier of MGFC
[(1_ MGFC),x] is non empty set
{(1_ MGFC),x} is non empty finite set
{(1_ MGFC)} is non empty trivial finite 1 -element set
{{(1_ MGFC),x},{(1_ MGFC)}} is non empty finite V44() set
the multF of MGFC . [(1_ MGFC),x] is set
x " is right_complementable Element of the carrier of MGFC
lc is Element of the carrier of p1
i * lc is Element of the carrier of p1
the multF of p1 . (i,lc) is Element of the carrier of p1
[i,lc] is non empty set
{i,lc} is non empty finite set
{{i,lc},{i}} is non empty finite V44() set
the multF of p1 . [i,lc] is set
lc * i is Element of the carrier of p1
the multF of p1 . (lc,i) is Element of the carrier of p1
[lc,i] is non empty set
{lc,i} is non empty finite set
{lc} is non empty trivial finite 1 -element set
{{lc,i},{lc}} is non empty finite V44() set
the multF of p1 . [lc,i] is set
x * (x ") is right_complementable Element of the carrier of MGFC
the multF of MGFC . (x,(x ")) is right_complementable Element of the carrier of MGFC
[x,(x ")] is non empty set
{x,(x ")} is non empty finite set
{{x,(x ")},{x}} is non empty finite V44() set
the multF of MGFC . [x,(x ")] is set
(x ") * x is right_complementable Element of the carrier of MGFC
the multF of MGFC . ((x "),x) is right_complementable Element of the carrier of MGFC
[(x "),x] is non empty set
{(x "),x} is non empty finite set
{(x ")} is non empty trivial finite 1 -element set
{{(x "),x},{(x ")}} is non empty finite V44() set
the multF of MGFC . [(x "),x] is set
qi is Element of the carrier of p1
i is Element of the carrier of p1
qi * i is Element of the carrier of p1
the multF of p1 . (qi,i) is Element of the carrier of p1
[qi,i] is non empty set
{qi,i} is non empty finite set
{qi} is non empty trivial finite 1 -element set
{{qi,i},{qi}} is non empty finite V44() set
the multF of p1 . [qi,i] is set
x is Element of the carrier of p1
(qi * i) * x is Element of the carrier of p1
the multF of p1 . ((qi * i),x) is Element of the carrier of p1
[(qi * i),x] is non empty set
{(qi * i),x} is non empty finite set
{(qi * i)} is non empty trivial finite 1 -element set
{{(qi * i),x},{(qi * i)}} is non empty finite V44() set
the multF of p1 . [(qi * i),x] is set
i * x is Element of the carrier of p1
the multF of p1 . (i,x) is Element of the carrier of p1
[i,x] is non empty set
{i,x} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,x},{i}} is non empty finite V44() set
the multF of p1 . [i,x] is set
qi * (i * x) is Element of the carrier of p1
the multF of p1 . (qi,(i * x)) is Element of the carrier of p1
[qi,(i * x)] is non empty set
{qi,(i * x)} is non empty finite set
{{qi,(i * x)},{qi}} is non empty finite V44() set
the multF of p1 . [qi,(i * x)] is set
mc is right_complementable Element of the carrier of MGFC
jcf is right_complementable Element of the carrier of MGFC
mc * jcf is right_complementable Element of the carrier of MGFC
the multF of MGFC . (mc,jcf) is right_complementable Element of the carrier of MGFC
[mc,jcf] is non empty set
{mc,jcf} is non empty finite set
{mc} is non empty trivial finite 1 -element set
{{mc,jcf},{mc}} is non empty finite V44() set
the multF of MGFC . [mc,jcf] is set
lc is right_complementable Element of the carrier of MGFC
lc * mc is right_complementable Element of the carrier of MGFC
the multF of MGFC . (lc,mc) is right_complementable Element of the carrier of MGFC
[lc,mc] is non empty set
{lc,mc} is non empty finite set
{lc} is non empty trivial finite 1 -element set
{{lc,mc},{lc}} is non empty finite V44() set
the multF of MGFC . [lc,mc] is set
(lc * mc) * jcf is right_complementable Element of the carrier of MGFC
the multF of MGFC . ((lc * mc),jcf) is right_complementable Element of the carrier of MGFC
[(lc * mc),jcf] is non empty set
{(lc * mc),jcf} is non empty finite set
{(lc * mc)} is non empty trivial finite 1 -element set
{{(lc * mc),jcf},{(lc * mc)}} is non empty finite V44() set
the multF of MGFC . [(lc * mc),jcf] is set
lc * (mc * jcf) is right_complementable Element of the carrier of MGFC
the multF of MGFC . (lc,(mc * jcf)) is right_complementable Element of the carrier of MGFC
[lc,(mc * jcf)] is non empty set
{lc,(mc * jcf)} is non empty finite set
{{lc,(mc * jcf)},{lc}} is non empty finite V44() set
the multF of MGFC . [lc,(mc * jcf)] is set
cMGFC is non empty strict unital Group-like associative multMagma
the carrier of cMGFC is non empty set
the multF of cMGFC is Relation-like [: the carrier of cMGFC, the carrier of cMGFC:] -defined the carrier of cMGFC -valued Function-like V25([: the carrier of cMGFC, the carrier of cMGFC:], the carrier of cMGFC) associative V51( the carrier of cMGFC) Element of bool [:[: the carrier of cMGFC, the carrier of cMGFC:], the carrier of cMGFC:]
[: the carrier of cMGFC, the carrier of cMGFC:] is Relation-like set
[:[: the carrier of cMGFC, the carrier of cMGFC:], the carrier of cMGFC:] is Relation-like set
bool [:[: the carrier of cMGFC, the carrier of cMGFC:], the carrier of cMGFC:] is set
the multF of MGFC || the carrier of cMGFC is Relation-like Function-like set
the multF of MGFC | [: the carrier of cMGFC, the carrier of cMGFC:] is Relation-like [: the carrier of cMGFC, the carrier of cMGFC:] -defined [: the carrier of MGFC, the carrier of MGFC:] -defined the carrier of MGFC -valued Function-like set
n is non empty strict unital Group-like associative multMagma
the carrier of n is non empty set
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like V25([: the carrier of n, the carrier of n:], the carrier of n) associative V51( the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the multF of MGFC || the carrier of n is Relation-like Function-like set
the multF of MGFC | [: the carrier of n, the carrier of n:] is Relation-like [: the carrier of n, the carrier of n:] -defined [: the carrier of MGFC, the carrier of MGFC:] -defined the carrier of MGFC -valued Function-like set
MGFC is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() doubleLoopStr
the carrier of MGFC is non empty non trivial set
(MGFC) is non empty strict unital Group-like associative multMagma
the carrier of (MGFC) is non empty set
0. MGFC is zero right_complementable Element of the carrier of MGFC
the ZeroF of MGFC is right_complementable Element of the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element Element of bool the carrier of MGFC
bool the carrier of MGFC is set
the carrier of (MGFC) \/ {(0. MGFC)} is non empty set
NonZero MGFC is non empty Element of bool the carrier of MGFC
[#] MGFC is non empty non proper Element of bool the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element set
([#] MGFC) \ {(0. MGFC)} is Element of bool the carrier of MGFC
MGFC is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() doubleLoopStr
the carrier of MGFC is non empty non trivial set
(MGFC) is non empty strict unital Group-like associative multMagma
the carrier of (MGFC) is non empty set
cMGFC is right_complementable Element of the carrier of MGFC
n is right_complementable Element of the carrier of MGFC
cMGFC * n is right_complementable Element of the carrier of MGFC
the multF of MGFC is Relation-like [: the carrier of MGFC, the carrier of MGFC:] -defined the carrier of MGFC -valued Function-like V25([: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC) associative Element of bool [:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:]
[: the carrier of MGFC, the carrier of MGFC:] is Relation-like set
[:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:] is Relation-like set
bool [:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:] is set
the multF of MGFC . (cMGFC,n) is right_complementable Element of the carrier of MGFC
[cMGFC,n] is non empty set
{cMGFC,n} is non empty finite set
{cMGFC} is non empty trivial finite 1 -element set
{{cMGFC,n},{cMGFC}} is non empty finite V44() set
the multF of MGFC . [cMGFC,n] is set
S is Element of the carrier of (MGFC)
fs is Element of the carrier of (MGFC)
S * fs is Element of the carrier of (MGFC)
the multF of (MGFC) is Relation-like [: the carrier of (MGFC), the carrier of (MGFC):] -defined the carrier of (MGFC) -valued Function-like V25([: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC)) associative V51( the carrier of (MGFC)) Element of bool [:[: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC):]
[: the carrier of (MGFC), the carrier of (MGFC):] is Relation-like set
[:[: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC):] is Relation-like set
bool [:[: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC):] is set
the multF of (MGFC) . (S,fs) is Element of the carrier of (MGFC)
[S,fs] is non empty set
{S,fs} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,fs},{S}} is non empty finite V44() set
the multF of (MGFC) . [S,fs] is set
[S,fs] is non empty Element of [: the carrier of (MGFC), the carrier of (MGFC):]
the multF of MGFC || the carrier of (MGFC) is Relation-like Function-like set
the multF of MGFC | [: the carrier of (MGFC), the carrier of (MGFC):] is Relation-like [: the carrier of (MGFC), the carrier of (MGFC):] -defined [: the carrier of MGFC, the carrier of MGFC:] -defined the carrier of MGFC -valued Function-like set
( the multF of MGFC || the carrier of (MGFC)) . (S,fs) is set
( the multF of MGFC || the carrier of (MGFC)) . [S,fs] is set
MGFC is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() doubleLoopStr
1_ MGFC is right_complementable Element of the carrier of MGFC
the carrier of MGFC is non empty non trivial set
1. MGFC is non zero right_complementable Element of the carrier of MGFC
the OneF of MGFC is right_complementable Element of the carrier of MGFC
(MGFC) is non empty strict unital Group-like associative multMagma
1_ (MGFC) is non being_of_order_0 Element of the carrier of (MGFC)
the carrier of (MGFC) is non empty set
NonZero MGFC is non empty Element of bool the carrier of MGFC
bool the carrier of MGFC is set
[#] MGFC is non empty non proper Element of bool the carrier of MGFC
0. MGFC is zero right_complementable Element of the carrier of MGFC
the ZeroF of MGFC is right_complementable Element of the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element set
([#] MGFC) \ {(0. MGFC)} is Element of bool the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element Element of bool the carrier of MGFC
n is Element of the carrier of (MGFC)
S is right_complementable Element of the carrier of MGFC
S * (1_ MGFC) is right_complementable Element of the carrier of MGFC
the multF of MGFC is Relation-like [: the carrier of MGFC, the carrier of MGFC:] -defined the carrier of MGFC -valued Function-like V25([: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC) associative Element of bool [:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:]
[: the carrier of MGFC, the carrier of MGFC:] is Relation-like set
[:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:] is Relation-like set
bool [:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:] is set
the multF of MGFC . (S,(1_ MGFC)) is right_complementable Element of the carrier of MGFC
[S,(1_ MGFC)] is non empty set
{S,(1_ MGFC)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(1_ MGFC)},{S}} is non empty finite V44() set
the multF of MGFC . [S,(1_ MGFC)] is set
cMGFC is Element of the carrier of (MGFC)
n * cMGFC is Element of the carrier of (MGFC)
the multF of (MGFC) is Relation-like [: the carrier of (MGFC), the carrier of (MGFC):] -defined the carrier of (MGFC) -valued Function-like V25([: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC)) associative V51( the carrier of (MGFC)) Element of bool [:[: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC):]
[: the carrier of (MGFC), the carrier of (MGFC):] is Relation-like set
[:[: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC):] is Relation-like set
bool [:[: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC):] is set
the multF of (MGFC) . (n,cMGFC) is Element of the carrier of (MGFC)
[n,cMGFC] is non empty set
{n,cMGFC} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,cMGFC},{n}} is non empty finite V44() set
the multF of (MGFC) . [n,cMGFC] is set
(1_ MGFC) * S is right_complementable Element of the carrier of MGFC
the multF of MGFC . ((1_ MGFC),S) is right_complementable Element of the carrier of MGFC
[(1_ MGFC),S] is non empty set
{(1_ MGFC),S} is non empty finite set
{(1_ MGFC)} is non empty trivial finite 1 -element set
{{(1_ MGFC),S},{(1_ MGFC)}} is non empty finite V44() set
the multF of MGFC . [(1_ MGFC),S] is set
cMGFC * n is Element of the carrier of (MGFC)
the multF of (MGFC) . (cMGFC,n) is Element of the carrier of (MGFC)
[cMGFC,n] is non empty set
{cMGFC,n} is non empty finite set
{cMGFC} is non empty trivial finite 1 -element set
{{cMGFC,n},{cMGFC}} is non empty finite V44() set
the multF of (MGFC) . [cMGFC,n] is set
MGFC is non empty non degenerated non trivial finite right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() doubleLoopStr
(MGFC) is non empty strict unital Group-like associative multMagma
the carrier of (MGFC) is non empty set
NonZero MGFC is non empty finite Element of bool the carrier of MGFC
the carrier of MGFC is non empty non trivial finite set
bool the carrier of MGFC is finite V44() set
[#] MGFC is non empty non proper finite Element of bool the carrier of MGFC
0. MGFC is zero right_complementable Element of the carrier of MGFC
the ZeroF of MGFC is right_complementable Element of the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element set
([#] MGFC) \ {(0. MGFC)} is finite Element of bool the carrier of MGFC
MGFC is non empty non degenerated non trivial finite right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() doubleLoopStr
(MGFC) is non empty finite strict unital Group-like associative multMagma
card (MGFC) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
the carrier of (MGFC) is non empty finite set
card the carrier of (MGFC) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal set
card MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
the carrier of MGFC is non empty non trivial finite set
card the carrier of MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal set
(card MGFC) - 1 is V31() real ext-real integer rational Element of INT
NonZero MGFC is non empty finite Element of bool the carrier of MGFC
bool the carrier of MGFC is finite V44() set
[#] MGFC is non empty non proper finite Element of bool the carrier of MGFC
0. MGFC is zero right_complementable Element of the carrier of MGFC
the ZeroF of MGFC is right_complementable Element of the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element set
([#] MGFC) \ {(0. MGFC)} is finite Element of bool the carrier of MGFC
card the carrier of (MGFC) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{(0. MGFC)} is non empty trivial finite 1 -element Element of bool the carrier of MGFC
card {(0. MGFC)} is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(card MGFC) - (card {(0. MGFC)}) is V31() real ext-real integer rational Element of INT
MGFC is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() doubleLoopStr
(MGFC) is non empty strict unital Group-like associative multMagma
the carrier of (MGFC) is non empty set
the carrier of MGFC is non empty non trivial set
cMGFC is set
NonZero MGFC is non empty Element of bool the carrier of MGFC
bool the carrier of MGFC is set
[#] MGFC is non empty non proper Element of bool the carrier of MGFC
0. MGFC is zero right_complementable Element of the carrier of MGFC
the ZeroF of MGFC is right_complementable Element of the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element set
([#] MGFC) \ {(0. MGFC)} is Element of bool the carrier of MGFC
MGFC is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() doubleLoopStr
(MGFC) is non empty strict unital Group-like associative multMagma
the carrier of (MGFC) is non empty set
the carrier of MGFC is non empty non trivial set
cMGFC is set
NonZero MGFC is non empty Element of bool the carrier of MGFC
bool the carrier of MGFC is set
[#] MGFC is non empty non proper Element of bool the carrier of MGFC
0. MGFC is zero right_complementable Element of the carrier of MGFC
the ZeroF of MGFC is right_complementable Element of the carrier of MGFC
{(0. MGFC)} is non empty trivial finite 1 -element set
([#] MGFC) \ {(0. MGFC)} is Element of bool the carrier of MGFC
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
n is set
S is V31() right_complementable Element of the carrier of F_Complex
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
cMGFC is V31() right_complementable Element of the carrier of F_Complex
n is V31() right_complementable Element of the carrier of F_Complex
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
(power F_Complex) . ((1_ F_Complex),MGFC) is V31() right_complementable Element of the carrier of F_Complex
[(1_ F_Complex),MGFC] is non empty set
{(1_ F_Complex),MGFC} is non empty finite V67() set
{(1_ F_Complex)} is non empty trivial finite 1 -element V67() set
{{(1_ F_Complex),MGFC},{(1_ F_Complex)}} is non empty finite V44() set
(power F_Complex) . [(1_ F_Complex),MGFC] is set
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
cMGFC is V31() right_complementable Element of the carrier of F_Complex
|.cMGFC.| is V31() real ext-real Element of REAL
Re cMGFC is V31() real ext-real Element of REAL
(Re cMGFC) ^2 is V31() real ext-real Element of REAL
K104((Re cMGFC),(Re cMGFC)) is V31() real ext-real set
Im cMGFC is V31() real ext-real Element of REAL
(Im cMGFC) ^2 is V31() real ext-real Element of REAL
K104((Im cMGFC),(Im cMGFC)) is V31() real ext-real set
((Re cMGFC) ^2) + ((Im cMGFC) ^2) is V31() real ext-real Element of REAL
sqrt (((Re cMGFC) ^2) + ((Im cMGFC) ^2)) is V31() real ext-real Element of REAL
(power F_Complex) . (cMGFC,MGFC) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,MGFC] is non empty set
{cMGFC,MGFC} is non empty finite V67() set
{cMGFC} is non empty trivial finite 1 -element V67() set
{{cMGFC,MGFC},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,MGFC] is set
|.cMGFC.| to_power MGFC is V31() real ext-real Element of REAL
|.cMGFC.| |^ MGFC is set
n is rational set
|.cMGFC.| #Q n is V31() real ext-real Element of REAL
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
cMGFC is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (cMGFC,MGFC) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,MGFC] is non empty set
{cMGFC,MGFC} is non empty finite V67() set
{cMGFC} is non empty trivial finite 1 -element V67() set
{{cMGFC,MGFC},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,MGFC] is set
Arg cMGFC is V31() real ext-real Element of REAL
- 1 is non empty V31() real ext-real non positive negative integer rational Element of INT
0 + (- 1) is non empty V31() real ext-real non positive negative integer rational Element of INT
MGFC * (Arg cMGFC) is V31() real ext-real Element of REAL
(MGFC * (Arg cMGFC)) / (2 * PI) is V31() real ext-real Element of REAL
((MGFC * (Arg cMGFC)) / (2 * PI)) + (- 1) is V31() real ext-real Element of REAL
|.cMGFC.| is V31() real ext-real Element of REAL
Re cMGFC is V31() real ext-real Element of REAL
(Re cMGFC) ^2 is V31() real ext-real Element of REAL
K104((Re cMGFC),(Re cMGFC)) is V31() real ext-real set
Im cMGFC is V31() real ext-real Element of REAL
(Im cMGFC) ^2 is V31() real ext-real Element of REAL
K104((Im cMGFC),(Im cMGFC)) is V31() real ext-real set
((Re cMGFC) ^2) + ((Im cMGFC) ^2) is V31() real ext-real Element of REAL
sqrt (((Re cMGFC) ^2) + ((Im cMGFC) ^2)) is V31() real ext-real Element of REAL
|.cMGFC.| to_power MGFC is V31() real ext-real Element of REAL
|.cMGFC.| |^ MGFC is set
n is rational set
|.cMGFC.| #Q n is V31() real ext-real Element of REAL
[\((MGFC * (Arg cMGFC)) / (2 * PI))/] is V31() real ext-real integer set
- [\((MGFC * (Arg cMGFC)) / (2 * PI))/] is V31() real ext-real integer rational Element of INT
(2 * PI) * (- [\((MGFC * (Arg cMGFC)) / (2 * PI))/]) is V31() real ext-real Element of REAL
((2 * PI) * (- [\((MGFC * (Arg cMGFC)) / (2 * PI))/])) + (MGFC * (Arg cMGFC)) is V31() real ext-real Element of REAL
fs is V31() real ext-real set
((MGFC * (Arg cMGFC)) / (2 * PI)) - 1 is V31() real ext-real Element of REAL
(- 1) + 1 is V31() real ext-real integer rational Element of INT
cos (MGFC * (Arg cMGFC)) is V31() real ext-real Element of REAL
q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * q is V31() real ext-real Element of REAL
((2 * PI) * q) + fs is V31() real ext-real Element of REAL
cos . (((2 * PI) * q) + fs) is V31() real ext-real Element of REAL
cos . fs is V31() real ext-real set
cos fs is set
sin (MGFC * (Arg cMGFC)) is V31() real ext-real Element of REAL
sin . (((2 * PI) * q) + fs) is V31() real ext-real Element of REAL
sin . fs is V31() real ext-real set
sin fs is set
S is V31() Element of COMPLEX
S |^ MGFC is set
|.S.| is V31() real ext-real Element of REAL
Re S is V31() real ext-real Element of REAL
(Re S) ^2 is V31() real ext-real Element of REAL
K104((Re S),(Re S)) is V31() real ext-real set
Im S is V31() real ext-real Element of REAL
(Im S) ^2 is V31() real ext-real Element of REAL
K104((Im S),(Im S)) is V31() real ext-real set
((Re S) ^2) + ((Im S) ^2) is V31() real ext-real Element of REAL
sqrt (((Re S) ^2) + ((Im S) ^2)) is V31() real ext-real Element of REAL
|.S.| |^ MGFC is V31() real ext-real Element of REAL
Arg S is V31() real ext-real Element of REAL
MGFC * (Arg S) is V31() real ext-real Element of REAL
cos (MGFC * (Arg S)) is V31() real ext-real Element of REAL
(|.S.| |^ MGFC) * (cos (MGFC * (Arg S))) is V31() real ext-real Element of REAL
sin (MGFC * (Arg S)) is V31() real ext-real Element of REAL
(|.S.| |^ MGFC) * (sin (MGFC * (Arg S))) is V31() real ext-real Element of REAL
((|.S.| |^ MGFC) * (sin (MGFC * (Arg S)))) * <i> is V31() Element of COMPLEX
((|.S.| |^ MGFC) * (cos (MGFC * (Arg S)))) + (((|.S.| |^ MGFC) * (sin (MGFC * (Arg S)))) * <i>) is V31() Element of COMPLEX
MGFC * 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC * (Arg cMGFC)) / (MGFC * 1) is V31() real ext-real Element of REAL
((2 * PI) * q) / MGFC is V31() real ext-real Element of REAL
MGFC / MGFC is non empty V31() real ext-real positive non negative rational Element of RAT
(MGFC / MGFC) * (Arg cMGFC) is V31() real ext-real Element of REAL
((MGFC / MGFC) * (Arg cMGFC)) / 1 is V31() real ext-real Element of REAL
(Arg cMGFC) / 1 is V31() real ext-real Element of REAL
cos (Arg cMGFC) is V31() real ext-real Element of REAL
|.cMGFC.| * (cos (Arg cMGFC)) is V31() real ext-real Element of REAL
sin (Arg cMGFC) is V31() real ext-real Element of REAL
|.cMGFC.| * (sin (Arg cMGFC)) is V31() real ext-real Element of REAL
[**(|.cMGFC.| * (cos (Arg cMGFC))),(|.cMGFC.| * (sin (Arg cMGFC)))**] is V31() right_complementable Element of the carrier of F_Complex
K104((|.cMGFC.| * (sin (Arg cMGFC))),<i>) is V31() set
K103((|.cMGFC.| * (cos (Arg cMGFC))),K104((|.cMGFC.| * (sin (Arg cMGFC))),<i>)) is V31() set
Arg (1_ F_Complex) is V31() real ext-real Element of REAL
fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * fs is V31() real ext-real Element of REAL
((2 * PI) * fs) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * fs) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * fs) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * fs) / MGFC)),(sin (((2 * PI) * fs) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * fs) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * fs) / MGFC)),K104((sin (((2 * PI) * fs) / MGFC)),<i>)) is V31() set
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC -root 1 is V31() real ext-real Element of REAL
n is V31() Element of COMPLEX
|.n.| is V31() real ext-real Element of REAL
Re n is V31() real ext-real Element of REAL
(Re n) ^2 is V31() real ext-real Element of REAL
K104((Re n),(Re n)) is V31() real ext-real set
Im n is V31() real ext-real Element of REAL
(Im n) ^2 is V31() real ext-real Element of REAL
K104((Im n),(Im n)) is V31() real ext-real set
((Re n) ^2) + ((Im n) ^2) is V31() real ext-real Element of REAL
sqrt (((Re n) ^2) + ((Im n) ^2)) is V31() real ext-real Element of REAL
MGFC -root |.n.| is V31() real ext-real Element of REAL
(Arg (1_ F_Complex)) + ((2 * PI) * fs) is V31() real ext-real Element of REAL
((Arg (1_ F_Complex)) + ((2 * PI) * fs)) / MGFC is V31() real ext-real Element of REAL
cos (((Arg (1_ F_Complex)) + ((2 * PI) * fs)) / MGFC) is V31() real ext-real Element of REAL
(MGFC -root |.n.|) * (cos (((Arg (1_ F_Complex)) + ((2 * PI) * fs)) / MGFC)) is V31() real ext-real Element of REAL
sin (((Arg (1_ F_Complex)) + ((2 * PI) * fs)) / MGFC) is V31() real ext-real Element of REAL
(MGFC -root |.n.|) * (sin (((Arg (1_ F_Complex)) + ((2 * PI) * fs)) / MGFC)) is V31() real ext-real Element of REAL
((MGFC -root |.n.|) * (sin (((Arg (1_ F_Complex)) + ((2 * PI) * fs)) / MGFC))) * <i> is V31() Element of COMPLEX
((MGFC -root |.n.|) * (cos (((Arg (1_ F_Complex)) + ((2 * PI) * fs)) / MGFC))) + (((MGFC -root |.n.|) * (sin (((Arg (1_ F_Complex)) + ((2 * PI) * fs)) / MGFC))) * <i>) is V31() Element of COMPLEX
n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * n is V31() real ext-real Element of REAL
((2 * PI) * n) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * n) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * n) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * n) / MGFC)),(sin (((2 * PI) * n) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * n) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * n) / MGFC)),K104((sin (((2 * PI) * n) / MGFC)),<i>)) is V31() set
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
cMGFC is V31() Element of COMPLEX
n is V31() Element of COMPLEX
cMGFC * n is V31() Element of COMPLEX
S is V31() right_complementable Element of the carrier of F_Complex
fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * fs is V31() real ext-real Element of REAL
((2 * PI) * fs) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * fs) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * fs) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * fs) / MGFC)),(sin (((2 * PI) * fs) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * fs) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * fs) / MGFC)),K104((sin (((2 * PI) * fs) / MGFC)),<i>)) is V31() set
q is V31() right_complementable Element of the carrier of F_Complex
qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * qc is V31() real ext-real Element of REAL
((2 * PI) * qc) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * qc) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * qc) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * qc) / MGFC)),(sin (((2 * PI) * qc) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * qc) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * qc) / MGFC)),K104((sin (((2 * PI) * qc) / MGFC)),<i>)) is V31() set
S * q is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) associative Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
the multF of F_Complex . (S,q) is V31() right_complementable Element of the carrier of F_Complex
[S,q] is non empty set
{S,q} is non empty finite V67() set
{S} is non empty trivial finite 1 -element V67() set
{{S,q},{S}} is non empty finite V44() set
the multF of F_Complex . [S,q] is set
S * q is V31() Element of COMPLEX
fs + qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(fs + qc) mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * ((fs + qc) mod MGFC) is V31() real ext-real Element of REAL
((2 * PI) * ((fs + qc) mod MGFC)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * ((fs + qc) mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * ((fs + qc) mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * ((fs + qc) mod MGFC)) / MGFC)),(sin (((2 * PI) * ((fs + qc) mod MGFC)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * ((fs + qc) mod MGFC)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * ((fs + qc) mod MGFC)) / MGFC)),K104((sin (((2 * PI) * ((fs + qc) mod MGFC)) / MGFC)),<i>)) is V31() set
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
{ [**(cos (((2 * PI) * b₁) / MGFC)),(sin (((2 * PI) * b₁) / MGFC))**] where b₁ is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT : not MGFC <= b₁ } is set
n is set
S is V31() right_complementable Element of the carrier of F_Complex
fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * fs is V31() real ext-real Element of REAL
((2 * PI) * fs) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * fs) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * fs) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * fs) / MGFC)),(sin (((2 * PI) * fs) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * fs) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * fs) / MGFC)),K104((sin (((2 * PI) * fs) / MGFC)),<i>)) is V31() set
fs mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * (fs mod MGFC) is V31() real ext-real Element of REAL
((2 * PI) * (fs mod MGFC)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * (fs mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * (fs mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * (fs mod MGFC)) / MGFC)),(sin (((2 * PI) * (fs mod MGFC)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * (fs mod MGFC)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * (fs mod MGFC)) / MGFC)),K104((sin (((2 * PI) * (fs mod MGFC)) / MGFC)),<i>)) is V31() set
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * S is V31() real ext-real Element of REAL
((2 * PI) * S) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * S) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * S) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * S) / MGFC)),(sin (((2 * PI) * S) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * S) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * S) / MGFC)),K104((sin (((2 * PI) * S) / MGFC)),<i>)) is V31() set
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
card (MGFC) is ordinal cardinal set
{ [**(cos (((2 * PI) * b₁) / MGFC)),(sin (((2 * PI) * b₁) / MGFC))**] where b₁ is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT : not MGFC <= b₁ } is set
(2 * PI) * 0 is V31() real ext-real Element of REAL
((2 * PI) * 0) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * 0) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * 0) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * 0) / MGFC)),(sin (((2 * PI) * 0) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * 0) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * 0) / MGFC)),K104((sin (((2 * PI) * 0) / MGFC)),<i>)) is V31() set
Seg MGFC is non empty finite MGFC -element V67() V68() V69() V70() V71() V72() Element of bool NAT
n is non empty set
S is set
fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC + 1) - 1 is V31() real ext-real integer rational Element of INT
fs - 1 is V31() real ext-real integer rational Element of INT
fs -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * q is V31() real ext-real Element of REAL
((2 * PI) * q) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * q) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * q) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * q) / MGFC)),(sin (((2 * PI) * q) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * q) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * q) / MGFC)),K104((sin (((2 * PI) * q) / MGFC)),<i>)) is V31() set
[:(Seg MGFC),n:] is Relation-like set
bool [:(Seg MGFC),n:] is set
S is Relation-like Seg MGFC -defined n -valued Function-like V25( Seg MGFC,n) finite finite-support Element of bool [:(Seg MGFC),n:]
fs is set
q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * q is V31() real ext-real Element of REAL
((2 * PI) * q) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * q) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * q) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * q) / MGFC)),(sin (((2 * PI) * q) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * q) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * q) / MGFC)),K104((sin (((2 * PI) * q) / MGFC)),<i>)) is V31() set
q + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(q + 1) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S . (q + 1) is set
qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qc -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * (qc -' 1) is V31() real ext-real Element of REAL
((2 * PI) * (qc -' 1)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * (qc -' 1)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * (qc -' 1)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * (qc -' 1)) / MGFC)),(sin (((2 * PI) * (qc -' 1)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * (qc -' 1)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * (qc -' 1)) / MGFC)),K104((sin (((2 * PI) * (qc -' 1)) / MGFC)),<i>)) is V31() set
rng S is finite set
dom S is finite set
fs is set
q is set
S . fs is set
S . q is set
qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qc -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * (qc -' 1) is V31() real ext-real Element of REAL
((2 * PI) * (qc -' 1)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * (qc -' 1)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * (qc -' 1)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * (qc -' 1)) / MGFC)),(sin (((2 * PI) * (qc -' 1)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * (qc -' 1)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * (qc -' 1)) / MGFC)),K104((sin (((2 * PI) * (qc -' 1)) / MGFC)),<i>)) is V31() set
qc - 1 is V31() real ext-real integer rational Element of INT
MGFC / MGFC is non empty V31() real ext-real positive non negative rational Element of RAT
(qc -' 1) / MGFC is V31() real ext-real non negative rational Element of RAT
1 * (2 * PI) is V31() real ext-real Element of REAL
((qc -' 1) / MGFC) * (2 * PI) is V31() real ext-real Element of REAL
ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
ps -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * (ps -' 1) is V31() real ext-real Element of REAL
((2 * PI) * (ps -' 1)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * (ps -' 1)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * (ps -' 1)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * (ps -' 1)) / MGFC)),(sin (((2 * PI) * (ps -' 1)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * (ps -' 1)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * (ps -' 1)) / MGFC)),K104((sin (((2 * PI) * (ps -' 1)) / MGFC)),<i>)) is V31() set
ps - 1 is V31() real ext-real integer rational Element of INT
(ps -' 1) / MGFC is V31() real ext-real non negative rational Element of RAT
((ps -' 1) / MGFC) * (2 * PI) is V31() real ext-real Element of REAL
(((2 * PI) * (qc -' 1)) / MGFC) * MGFC is V31() real ext-real Element of REAL
(ps -' 1) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S .: (Seg MGFC) is finite set
card (Seg MGFC) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
card { [**(cos (((2 * PI) * b₁) / MGFC)),(sin (((2 * PI) * b₁) / MGFC))**] where b₁ is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT : not MGFC <= b₁ } is ordinal cardinal set
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
card (MGFC) is ordinal non empty cardinal set
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of cMGFC, 1_ F_Complex } is set
(MGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
n is ordinal natural V31() real ext-real non negative integer finite cardinal set
cMGFC * n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs is set
q is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (q,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
[q,cMGFC] is non empty set
{q,cMGFC} is non empty finite V67() set
{q} is non empty trivial finite 1 -element V67() set
{{q,cMGFC},{q}} is non empty finite V44() set
(power F_Complex) . [q,cMGFC] is set
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power F_Complex) . (((power F_Complex) . (q,cMGFC)),S) is V31() right_complementable Element of the carrier of F_Complex
[((power F_Complex) . (q,cMGFC)),S] is non empty set
{((power F_Complex) . (q,cMGFC)),S} is non empty finite V67() set
{((power F_Complex) . (q,cMGFC))} is non empty trivial finite 1 -element V67() set
{{((power F_Complex) . (q,cMGFC)),S},{((power F_Complex) . (q,cMGFC))}} is non empty finite V44() set
(power F_Complex) . [((power F_Complex) . (q,cMGFC)),S] is set
(power F_Complex) . (q,MGFC) is V31() right_complementable Element of the carrier of F_Complex
[q,MGFC] is non empty set
{q,MGFC} is non empty finite V67() set
{{q,MGFC},{q}} is non empty finite V44() set
(power F_Complex) . [q,MGFC] is set
MGFC is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() doubleLoopStr
(MGFC) is non empty strict unital Group-like associative multMagma
the carrier of (MGFC) is non empty set
the carrier of MGFC is non empty non trivial set
power (MGFC) is Relation-like [: the carrier of (MGFC),NAT:] -defined the carrier of (MGFC) -valued Function-like V25([: the carrier of (MGFC),NAT:], the carrier of (MGFC)) Element of bool [:[: the carrier of (MGFC),NAT:], the carrier of (MGFC):]
[: the carrier of (MGFC),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() V60() set
[:[: the carrier of (MGFC),NAT:], the carrier of (MGFC):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (MGFC),NAT:], the carrier of (MGFC):] is non empty non trivial non finite set
power MGFC is Relation-like [: the carrier of MGFC,NAT:] -defined the carrier of MGFC -valued Function-like V25([: the carrier of MGFC,NAT:], the carrier of MGFC) Element of bool [:[: the carrier of MGFC,NAT:], the carrier of MGFC:]
[: the carrier of MGFC,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() V60() set
[:[: the carrier of MGFC,NAT:], the carrier of MGFC:] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of MGFC,NAT:], the carrier of MGFC:] is non empty non trivial non finite set
cMGFC is Element of the carrier of (MGFC)
n is right_complementable Element of the carrier of MGFC
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power (MGFC)) . (cMGFC,S) is Element of the carrier of (MGFC)
[cMGFC,S] is non empty set
{cMGFC,S} is non empty finite set
{cMGFC} is non empty trivial finite 1 -element set
{{cMGFC,S},{cMGFC}} is non empty finite V44() set
(power (MGFC)) . [cMGFC,S] is set
(power MGFC) . (n,S) is right_complementable Element of the carrier of MGFC
[n,S] is non empty set
{n,S} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,S},{n}} is non empty finite V44() set
(power MGFC) . [n,S] is set
S + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power (MGFC)) . (cMGFC,(S + 1)) is Element of the carrier of (MGFC)
[cMGFC,(S + 1)] is non empty set
{cMGFC,(S + 1)} is non empty finite set
{{cMGFC,(S + 1)},{cMGFC}} is non empty finite V44() set
(power (MGFC)) . [cMGFC,(S + 1)] is set
(power MGFC) . (n,(S + 1)) is right_complementable Element of the carrier of MGFC
[n,(S + 1)] is non empty set
{n,(S + 1)} is non empty finite set
{{n,(S + 1)},{n}} is non empty finite V44() set
(power MGFC) . [n,(S + 1)] is set
((power (MGFC)) . (cMGFC,S)) * cMGFC is Element of the carrier of (MGFC)
the multF of (MGFC) is Relation-like [: the carrier of (MGFC), the carrier of (MGFC):] -defined the carrier of (MGFC) -valued Function-like V25([: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC)) associative V51( the carrier of (MGFC)) Element of bool [:[: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC):]
[: the carrier of (MGFC), the carrier of (MGFC):] is Relation-like set
[:[: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC):] is Relation-like set
bool [:[: the carrier of (MGFC), the carrier of (MGFC):], the carrier of (MGFC):] is set
the multF of (MGFC) . (((power (MGFC)) . (cMGFC,S)),cMGFC) is Element of the carrier of (MGFC)
[((power (MGFC)) . (cMGFC,S)),cMGFC] is non empty set
{((power (MGFC)) . (cMGFC,S)),cMGFC} is non empty finite set
{((power (MGFC)) . (cMGFC,S))} is non empty trivial finite 1 -element set
{{((power (MGFC)) . (cMGFC,S)),cMGFC},{((power (MGFC)) . (cMGFC,S))}} is non empty finite V44() set
the multF of (MGFC) . [((power (MGFC)) . (cMGFC,S)),cMGFC] is set
((power MGFC) . (n,S)) * n is right_complementable Element of the carrier of MGFC
the multF of MGFC is Relation-like [: the carrier of MGFC, the carrier of MGFC:] -defined the carrier of MGFC -valued Function-like V25([: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC) associative Element of bool [:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:]
[: the carrier of MGFC, the carrier of MGFC:] is Relation-like set
[:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:] is Relation-like set
bool [:[: the carrier of MGFC, the carrier of MGFC:], the carrier of MGFC:] is set
the multF of MGFC . (((power MGFC) . (n,S)),n) is right_complementable Element of the carrier of MGFC
[((power MGFC) . (n,S)),n] is non empty set
{((power MGFC) . (n,S)),n} is non empty finite set
{((power MGFC) . (n,S))} is non empty trivial finite 1 -element set
{{((power MGFC) . (n,S)),n},{((power MGFC) . (n,S))}} is non empty finite V44() set
the multF of MGFC . [((power MGFC) . (n,S)),n] is set
(power (MGFC)) . (cMGFC,0) is Element of the carrier of (MGFC)
[cMGFC,0] is non empty set
{cMGFC,0} is non empty finite set
{cMGFC} is non empty trivial finite 1 -element set
{{cMGFC,0},{cMGFC}} is non empty finite V44() set
(power (MGFC)) . [cMGFC,0] is set
1_ (MGFC) is non being_of_order_0 Element of the carrier of (MGFC)
(power MGFC) . (n,0) is right_complementable Element of the carrier of MGFC
[n,0] is non empty set
{n,0} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,0},{n}} is non empty finite V44() set
(power MGFC) . [n,0] is set
1_ MGFC is right_complementable Element of the carrier of MGFC
1. MGFC is non zero right_complementable Element of the carrier of MGFC
the OneF of MGFC is right_complementable Element of the carrier of MGFC
(F_Complex) is non empty strict unital Group-like associative multMagma
the carrier of (F_Complex) is non empty set
S is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(S) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of S, 1_ F_Complex } is set
fs is Element of the carrier of (F_Complex)
q is V31() right_complementable Element of the carrier of F_Complex
1_ (F_Complex) is non being_of_order_0 Element of the carrier of (F_Complex)
(power F_Complex) . (q,S) is V31() right_complementable Element of the carrier of F_Complex
[q,S] is non empty set
{q,S} is non empty finite V67() set
{q} is non empty trivial finite 1 -element V67() set
{{q,S},{q}} is non empty finite V44() set
(power F_Complex) . [q,S] is set
fs |^ S is Element of the carrier of (F_Complex)
power (F_Complex) is Relation-like [: the carrier of (F_Complex),NAT:] -defined the carrier of (F_Complex) -valued Function-like V25([: the carrier of (F_Complex),NAT:], the carrier of (F_Complex)) Element of bool [:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):]
[: the carrier of (F_Complex),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() V60() set
[:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):] is non empty non trivial non finite set
(power (F_Complex)) . (fs,S) is set
[fs,S] is non empty set
{fs,S} is non empty finite set
{fs} is non empty trivial finite 1 -element set
{{fs,S},{fs}} is non empty finite V44() set
(power (F_Complex)) . [fs,S] is set
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * cMGFC is V31() real ext-real Element of REAL
((2 * PI) * cMGFC) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * cMGFC) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * cMGFC) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * cMGFC) / MGFC)),(sin (((2 * PI) * cMGFC) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * cMGFC) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * cMGFC) / MGFC)),K104((sin (((2 * PI) * cMGFC) / MGFC)),<i>)) is V31() set
cMGFC gcd MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC div (cMGFC gcd MGFC) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is ordinal natural V31() real ext-real non negative integer finite cardinal set
n * S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs is ordinal natural V31() real ext-real non negative integer finite cardinal set
n * fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q is Element of the carrier of (F_Complex)
ord q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
MGFC gcd cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC div n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n * 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered rational V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of NAT
MGFC mod n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n * 0) + (MGFC mod n) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n * p1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n * p1) + 0 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
1_ (F_Complex) is non being_of_order_0 Element of the carrier of (F_Complex)
ps is ordinal natural V31() real ext-real non negative integer finite cardinal set
q |^ ps is Element of the carrier of (F_Complex)
power (F_Complex) is Relation-like [: the carrier of (F_Complex),NAT:] -defined the carrier of (F_Complex) -valued Function-like V25([: the carrier of (F_Complex),NAT:], the carrier of (F_Complex)) Element of bool [:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):]
[: the carrier of (F_Complex),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() V60() set
[:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):] is non empty non trivial non finite set
(power (F_Complex)) . (q,ps) is set
[q,ps] is non empty set
{q,ps} is non empty finite set
{q} is non empty trivial finite 1 -element set
{{q,ps},{q}} is non empty finite V44() set
(power (F_Complex)) . [q,ps] is set
qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC * qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC * qi) mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
i is ordinal natural V31() real ext-real non negative integer finite cardinal set
MGFC * i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
x is V31() real ext-real integer set
n * x is V31() real ext-real integer rational Element of INT
lc is V31() real ext-real integer set
n * lc is V31() real ext-real integer rational Element of INT
qi * x is V31() real ext-real integer rational Element of INT
(qi * x) * n is V31() real ext-real integer rational Element of INT
lc * n is V31() real ext-real integer rational Element of INT
i * (lc * n) is V31() real ext-real integer rational Element of INT
i * lc is V31() real ext-real integer rational Element of INT
(i * lc) * n is V31() real ext-real integer rational Element of INT
((i * lc) * n) / n is V31() real ext-real rational Element of RAT
jcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
mc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qi * mc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(((2 * PI) * cMGFC) / MGFC) * qi is V31() real ext-real Element of REAL
MGFC / qi is V31() real ext-real non negative rational Element of RAT
((2 * PI) * cMGFC) / (MGFC / qi) is V31() real ext-real Element of REAL
((2 * PI) * cMGFC) * qi is V31() real ext-real Element of REAL
(((2 * PI) * cMGFC) * qi) / MGFC is V31() real ext-real Element of REAL
(2 * PI) * MGFC is V31() real ext-real Element of REAL
(2 * PI) * ((cMGFC * qi) mod MGFC) is V31() real ext-real Element of REAL
((2 * PI) * MGFC) / MGFC is V31() real ext-real Element of REAL
((2 * PI) * ((cMGFC * qi) mod MGFC)) / MGFC is V31() real ext-real Element of REAL
q |^ qi is Element of the carrier of (F_Complex)
(power (F_Complex)) . (q,qi) is set
[q,qi] is non empty set
{q,qi} is non empty finite set
{{q,qi},{q}} is non empty finite V44() set
(power (F_Complex)) . [q,qi] is set
qc is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (qc,qi) is V31() right_complementable Element of the carrier of F_Complex
[qc,qi] is non empty set
{qc,qi} is non empty finite V67() set
{qc} is non empty trivial finite 1 -element V67() set
{{qc,qi},{qc}} is non empty finite V44() set
(power F_Complex) . [qc,qi] is set
qc |^ qi is set
(2 * PI) * (cMGFC * qi) is V31() real ext-real Element of REAL
((2 * PI) * (cMGFC * qi)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * (cMGFC * qi)) / MGFC) is V31() real ext-real Element of REAL
sin ((((2 * PI) * cMGFC) * qi) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * (cMGFC * qi)) / MGFC)),(sin ((((2 * PI) * cMGFC) * qi) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin ((((2 * PI) * cMGFC) * qi) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * (cMGFC * qi)) / MGFC)),K104((sin ((((2 * PI) * cMGFC) * qi) / MGFC)),<i>)) is V31() set
cos (((2 * PI) * ((cMGFC * qi) mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * ((cMGFC * qi) mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * ((cMGFC * qi) mod MGFC)) / MGFC)),(sin (((2 * PI) * ((cMGFC * qi) mod MGFC)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * ((cMGFC * qi) mod MGFC)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * ((cMGFC * qi) mod MGFC)) / MGFC)),K104((sin (((2 * PI) * ((cMGFC * qi) mod MGFC)) / MGFC)),<i>)) is V31() set
(((2 * PI) * cMGFC) / MGFC) * p1 is V31() real ext-real Element of REAL
ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n * ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * (n * ps) is V31() real ext-real Element of REAL
MGFC / p1 is V31() real ext-real non negative rational Element of RAT
((2 * PI) * (n * ps)) / (MGFC / p1) is V31() real ext-real Element of REAL
((2 * PI) * (n * ps)) * p1 is V31() real ext-real Element of REAL
(((2 * PI) * (n * ps)) * p1) / MGFC is V31() real ext-real Element of REAL
(2 * PI) * ps is V31() real ext-real Element of REAL
((2 * PI) * ps) * MGFC is V31() real ext-real Element of REAL
(((2 * PI) * ps) * MGFC) / MGFC is V31() real ext-real Element of REAL
((2 * PI) * ps) + 0 is V31() real ext-real Element of REAL
q |^ (MGFC div n) is Element of the carrier of (F_Complex)
power (F_Complex) is Relation-like [: the carrier of (F_Complex),NAT:] -defined the carrier of (F_Complex) -valued Function-like V25([: the carrier of (F_Complex),NAT:], the carrier of (F_Complex)) Element of bool [:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):]
[: the carrier of (F_Complex),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() V60() set
[:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):] is non empty non trivial non finite set
(power (F_Complex)) . (q,(MGFC div n)) is set
[q,(MGFC div n)] is non empty set
{q,(MGFC div n)} is non empty finite set
{q} is non empty trivial finite 1 -element set
{{q,(MGFC div n)},{q}} is non empty finite V44() set
(power (F_Complex)) . [q,(MGFC div n)] is set
qc is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (qc,p1) is V31() right_complementable Element of the carrier of F_Complex
[qc,p1] is non empty set
{qc,p1} is non empty finite V67() set
{qc} is non empty trivial finite 1 -element V67() set
{{qc,p1},{qc}} is non empty finite V44() set
(power F_Complex) . [qc,p1] is set
qc |^ p1 is set
cos ((((2 * PI) * cMGFC) / MGFC) * p1) is V31() real ext-real Element of REAL
sin ((((2 * PI) * cMGFC) / MGFC) * p1) is V31() real ext-real Element of REAL
[**(cos ((((2 * PI) * cMGFC) / MGFC) * p1)),(sin ((((2 * PI) * cMGFC) / MGFC) * p1))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin ((((2 * PI) * cMGFC) / MGFC) * p1)),<i>) is V31() set
K103((cos ((((2 * PI) * cMGFC) / MGFC) * p1)),K104((sin ((((2 * PI) * cMGFC) / MGFC) * p1)),<i>)) is V31() set
sin (((2 * PI) * ps) + 0) is V31() real ext-real Element of REAL
[**(cos 0),(sin (((2 * PI) * ps) + 0))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * ps) + 0)),<i>) is V31() set
K103((cos 0),K104((sin (((2 * PI) * ps) + 0)),<i>)) is V31() set
0 * <i> is V31() Element of COMPLEX
1 + (0 * <i>) is V31() Element of COMPLEX
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
S is set
fs is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (fs,MGFC) is V31() right_complementable Element of the carrier of F_Complex
[fs,MGFC] is non empty set
{fs,MGFC} is non empty finite V67() set
{fs} is non empty trivial finite 1 -element V67() set
{{fs,MGFC},{fs}} is non empty finite V44() set
(power F_Complex) . [fs,MGFC] is set
{(0. F_Complex)} is non empty trivial finite 1 -element V67() Element of bool the carrier of F_Complex
NonZero F_Complex is non empty Element of bool the carrier of F_Complex
[#] F_Complex is non empty non proper Element of bool the carrier of F_Complex
{(0. F_Complex)} is non empty trivial finite 1 -element V67() set
([#] F_Complex) \ {(0. F_Complex)} is Element of bool the carrier of F_Complex
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * 1 is V31() real ext-real Element of REAL
((2 * PI) * 1) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * 1) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * 1) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * 1) / MGFC)),(sin (((2 * PI) * 1) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * 1) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * 1) / MGFC)),K104((sin (((2 * PI) * 1) / MGFC)),<i>)) is V31() set
(MGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
n is Element of the carrier of (F_Complex)
ord n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
1 gcd MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC div (1 gcd MGFC) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of cMGFC, 1_ F_Complex } is set
n is Element of the carrier of (F_Complex)
ord n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
1_ (F_Complex) is non being_of_order_0 Element of the carrier of (F_Complex)
q is ordinal natural V31() real ext-real non negative integer finite cardinal set
(ord n) * q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n |^ (ord n) is Element of the carrier of (F_Complex)
power (F_Complex) is Relation-like [: the carrier of (F_Complex),NAT:] -defined the carrier of (F_Complex) -valued Function-like V25([: the carrier of (F_Complex),NAT:], the carrier of (F_Complex)) Element of bool [:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):]
[: the carrier of (F_Complex),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V57() V58() V59() V60() set
[:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (F_Complex),NAT:], the carrier of (F_Complex):] is non empty non trivial non finite set
(power (F_Complex)) . (n,(ord n)) is set
[n,(ord n)] is non empty set
{n,(ord n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(ord n)},{n}} is non empty finite V44() set
(power (F_Complex)) . [n,(ord n)] is set
(n |^ (ord n)) |^ q is Element of the carrier of (F_Complex)
(power (F_Complex)) . ((n |^ (ord n)),q) is set
[(n |^ (ord n)),q] is non empty set
{(n |^ (ord n)),q} is non empty finite set
{(n |^ (ord n))} is non empty trivial finite 1 -element set
{{(n |^ (ord n)),q},{(n |^ (ord n))}} is non empty finite V44() set
(power (F_Complex)) . [(n |^ (ord n)),q] is set
n |^ cMGFC is Element of the carrier of (F_Complex)
(power (F_Complex)) . (n,cMGFC) is set
[n,cMGFC] is non empty set
{n,cMGFC} is non empty finite set
{{n,cMGFC},{n}} is non empty finite V44() set
(power (F_Complex)) . [n,cMGFC] is set
S is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (S,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
[S,cMGFC] is non empty set
{S,cMGFC} is non empty finite V67() set
{S} is non empty trivial finite 1 -element V67() set
{{S,cMGFC},{S}} is non empty finite V44() set
(power F_Complex) . [S,cMGFC] is set
q is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (q,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
[q,cMGFC] is non empty set
{q,cMGFC} is non empty finite V67() set
{q} is non empty trivial finite 1 -element V67() set
{{q,cMGFC},{q}} is non empty finite V44() set
(power F_Complex) . [q,cMGFC] is set
n is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of n, 1_ F_Complex } is set
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ divides n } is set
q is set
qc is Element of the carrier of (F_Complex)
ord qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qc is Element of the carrier of (F_Complex)
ord qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of n, 1_ F_Complex } is set
S is set
fs is Element of the carrier of (F_Complex)
ord fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs is Element of the carrier of (F_Complex)
ord fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
the multF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) associative Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is Relation-like set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is Relation-like set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
the multF of F_Complex || (MGFC) is Relation-like Function-like set
[:(MGFC),(MGFC):] is Relation-like finite set
the multF of F_Complex | [:(MGFC),(MGFC):] is Relation-like [:(MGFC),(MGFC):] -defined [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like finite finite-support set
dom multcomplex is Relation-like set
dom ( the multF of F_Complex || (MGFC)) is set
(dom multcomplex) /\ [:(MGFC),(MGFC):] is Relation-like finite set
S is set
( the multF of F_Complex || (MGFC)) . S is set
fs is set
q is set
[fs,q] is non empty set
{fs,q} is non empty finite set
{fs} is non empty trivial finite 1 -element set
{{fs,q},{fs}} is non empty finite V44() set
p1 is V31() Element of COMPLEX
qc is V31() Element of COMPLEX
multcomplex . (p1,qc) is V31() Element of COMPLEX
[p1,qc] is non empty set
{p1,qc} is non empty finite V67() set
{p1} is non empty trivial finite 1 -element V67() set
{{p1,qc},{p1}} is non empty finite V44() set
multcomplex . [p1,qc] is V31() set
p1 * qc is V31() Element of COMPLEX
[p1,qc] is non empty Element of [:COMPLEX,COMPLEX:]
( the multF of F_Complex || (MGFC)) . [p1,qc] is set
multcomplex . [p1,qc] is V31() Element of COMPLEX
[:[:(MGFC),(MGFC):],(MGFC):] is Relation-like finite set
bool [:[:(MGFC),(MGFC):],(MGFC):] is finite V44() set
S is Relation-like [:(MGFC),(MGFC):] -defined (MGFC) -valued Function-like V25([:(MGFC),(MGFC):],(MGFC)) finite finite-support Element of bool [:[:(MGFC),(MGFC):],(MGFC):]
multMagma(# (MGFC),S #) is non empty strict multMagma
the carrier of multMagma(# (MGFC),S #) is non empty set
(2 * PI) * 0 is V31() real ext-real Element of REAL
((2 * PI) * 0) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * 0) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * 0) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * 0) / MGFC)),(sin (((2 * PI) * 0) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * 0) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * 0) / MGFC)),K104((sin (((2 * PI) * 0) / MGFC)),<i>)) is V31() set
q is Element of the carrier of multMagma(# (MGFC),S #)
qc is Element of the carrier of multMagma(# (MGFC),S #)
qc * q is Element of the carrier of multMagma(# (MGFC),S #)
the multF of multMagma(# (MGFC),S #) is Relation-like [: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):] -defined the carrier of multMagma(# (MGFC),S #) -valued Function-like V25([: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):], the carrier of multMagma(# (MGFC),S #)) Element of bool [:[: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):], the carrier of multMagma(# (MGFC),S #):]
[: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):] is Relation-like set
[:[: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):], the carrier of multMagma(# (MGFC),S #):] is Relation-like set
bool [:[: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):], the carrier of multMagma(# (MGFC),S #):] is set
the multF of multMagma(# (MGFC),S #) . (qc,q) is Element of the carrier of multMagma(# (MGFC),S #)
[qc,q] is non empty set
{qc,q} is non empty finite set
{qc} is non empty trivial finite 1 -element set
{{qc,q},{qc}} is non empty finite V44() set
the multF of multMagma(# (MGFC),S #) . [qc,q] is set
q * qc is Element of the carrier of multMagma(# (MGFC),S #)
the multF of multMagma(# (MGFC),S #) . (q,qc) is Element of the carrier of multMagma(# (MGFC),S #)
[q,qc] is non empty set
{q,qc} is non empty finite set
{q} is non empty trivial finite 1 -element set
{{q,qc},{q}} is non empty finite V44() set
the multF of multMagma(# (MGFC),S #) . [q,qc] is set
p1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * p1 is V31() real ext-real Element of REAL
((2 * PI) * p1) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * p1) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * p1) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * p1) / MGFC)),(sin (((2 * PI) * p1) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * p1) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * p1) / MGFC)),K104((sin (((2 * PI) * p1) / MGFC)),<i>)) is V31() set
p1 mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
i is ordinal natural V31() real ext-real non negative integer finite cardinal set
(p1 mod MGFC) + i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 div MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC * (p1 div MGFC) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC * (p1 div MGFC)) + (p1 mod MGFC) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 + i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(p1 + i) mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(p1 div MGFC) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
((p1 div MGFC) + 1) * MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(((p1 div MGFC) + 1) * MGFC) mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 + qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(p1 + qi) mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 + qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(p1 + qi) mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * qi is V31() real ext-real Element of REAL
((2 * PI) * qi) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * qi) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * qi) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * qi) / MGFC)),(sin (((2 * PI) * qi) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * qi) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * qi) / MGFC)),K104((sin (((2 * PI) * qi) / MGFC)),<i>)) is V31() set
x is Element of the carrier of multMagma(# (MGFC),S #)
lc is V31() right_complementable Element of the carrier of F_Complex
ps is V31() right_complementable Element of the carrier of F_Complex
lc * ps is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . (lc,ps) is V31() right_complementable Element of the carrier of F_Complex
[lc,ps] is non empty set
{lc,ps} is non empty finite V67() set
{lc} is non empty trivial finite 1 -element V67() set
{{lc,ps},{lc}} is non empty finite V44() set
the multF of F_Complex . [lc,ps] is set
lc * ps is V31() Element of COMPLEX
qi + p1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(qi + p1) mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * ((qi + p1) mod MGFC) is V31() real ext-real Element of REAL
((2 * PI) * ((qi + p1) mod MGFC)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * ((qi + p1) mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * ((qi + p1) mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * ((qi + p1) mod MGFC)) / MGFC)),(sin (((2 * PI) * ((qi + p1) mod MGFC)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * ((qi + p1) mod MGFC)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * ((qi + p1) mod MGFC)) / MGFC)),K104((sin (((2 * PI) * ((qi + p1) mod MGFC)) / MGFC)),<i>)) is V31() set
[x,qc] is non empty Element of [: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):]
{x,qc} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,qc},{x}} is non empty finite V44() set
x * qc is Element of the carrier of multMagma(# (MGFC),S #)
the multF of multMagma(# (MGFC),S #) . (x,qc) is Element of the carrier of multMagma(# (MGFC),S #)
[x,qc] is non empty set
the multF of multMagma(# (MGFC),S #) . [x,qc] is set
ps * lc is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . (ps,lc) is V31() right_complementable Element of the carrier of F_Complex
[ps,lc] is non empty set
{ps,lc} is non empty finite V67() set
{ps} is non empty trivial finite 1 -element V67() set
{{ps,lc},{ps}} is non empty finite V44() set
the multF of F_Complex . [ps,lc] is set
ps * lc is V31() Element of COMPLEX
[qc,x] is non empty Element of [: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):]
{qc,x} is non empty finite set
{{qc,x},{qc}} is non empty finite V44() set
qc * x is Element of the carrier of multMagma(# (MGFC),S #)
the multF of multMagma(# (MGFC),S #) . (qc,x) is Element of the carrier of multMagma(# (MGFC),S #)
[qc,x] is non empty set
the multF of multMagma(# (MGFC),S #) . [qc,x] is set
[qc,q] is non empty Element of [: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):]
[q,qc] is non empty Element of [: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):]
p1 is V31() right_complementable Element of the carrier of F_Complex
ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * ps is V31() real ext-real Element of REAL
((2 * PI) * ps) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * ps) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * ps) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * ps) / MGFC)),(sin (((2 * PI) * ps) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * ps) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * ps) / MGFC)),K104((sin (((2 * PI) * ps) / MGFC)),<i>)) is V31() set
(1_ F_Complex) * p1 is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((1_ F_Complex),p1) is V31() right_complementable Element of the carrier of F_Complex
[(1_ F_Complex),p1] is non empty set
{(1_ F_Complex),p1} is non empty finite V67() set
{(1_ F_Complex)} is non empty trivial finite 1 -element V67() set
{{(1_ F_Complex),p1},{(1_ F_Complex)}} is non empty finite V44() set
the multF of F_Complex . [(1_ F_Complex),p1] is set
(1_ F_Complex) * p1 is V31() Element of COMPLEX
ps + 0 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(ps + 0) mod MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(2 * PI) * ((ps + 0) mod MGFC) is V31() real ext-real Element of REAL
((2 * PI) * ((ps + 0) mod MGFC)) / MGFC is V31() real ext-real Element of REAL
cos (((2 * PI) * ((ps + 0) mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
sin (((2 * PI) * ((ps + 0) mod MGFC)) / MGFC) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * ((ps + 0) mod MGFC)) / MGFC)),(sin (((2 * PI) * ((ps + 0) mod MGFC)) / MGFC))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * ((ps + 0) mod MGFC)) / MGFC)),<i>) is V31() set
K103((cos (((2 * PI) * ((ps + 0) mod MGFC)) / MGFC)),K104((sin (((2 * PI) * ((ps + 0) mod MGFC)) / MGFC)),<i>)) is V31() set
p1 * (1_ F_Complex) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . (p1,(1_ F_Complex)) is V31() right_complementable Element of the carrier of F_Complex
[p1,(1_ F_Complex)] is non empty set
{p1,(1_ F_Complex)} is non empty finite V67() set
{p1} is non empty trivial finite 1 -element V67() set
{{p1,(1_ F_Complex)},{p1}} is non empty finite V44() set
the multF of F_Complex . [p1,(1_ F_Complex)] is set
p1 * (1_ F_Complex) is V31() Element of COMPLEX
p1 is Element of the carrier of multMagma(# (MGFC),S #)
qc * p1 is Element of the carrier of multMagma(# (MGFC),S #)
the multF of multMagma(# (MGFC),S #) . (qc,p1) is Element of the carrier of multMagma(# (MGFC),S #)
[qc,p1] is non empty set
{qc,p1} is non empty finite set
{{qc,p1},{qc}} is non empty finite V44() set
the multF of multMagma(# (MGFC),S #) . [qc,p1] is set
p1 * qc is Element of the carrier of multMagma(# (MGFC),S #)
the multF of multMagma(# (MGFC),S #) . (p1,qc) is Element of the carrier of multMagma(# (MGFC),S #)
[p1,qc] is non empty set
{p1,qc} is non empty finite set
{p1} is non empty trivial finite 1 -element set
{{p1,qc},{p1}} is non empty finite V44() set
the multF of multMagma(# (MGFC),S #) . [p1,qc] is set
rng S is finite set
p1 is Element of the carrier of multMagma(# (MGFC),S #)
ps is Element of the carrier of multMagma(# (MGFC),S #)
p1 * ps is Element of the carrier of multMagma(# (MGFC),S #)
the multF of multMagma(# (MGFC),S #) is Relation-like [: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):] -defined the carrier of multMagma(# (MGFC),S #) -valued Function-like V25([: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):], the carrier of multMagma(# (MGFC),S #)) Element of bool [:[: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):], the carrier of multMagma(# (MGFC),S #):]
[: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):] is Relation-like set
[:[: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):], the carrier of multMagma(# (MGFC),S #):] is Relation-like set
bool [:[: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):], the carrier of multMagma(# (MGFC),S #):] is set
the multF of multMagma(# (MGFC),S #) . (p1,ps) is Element of the carrier of multMagma(# (MGFC),S #)
[p1,ps] is non empty set
{p1,ps} is non empty finite set
{p1} is non empty trivial finite 1 -element set
{{p1,ps},{p1}} is non empty finite V44() set
the multF of multMagma(# (MGFC),S #) . [p1,ps] is set
qi is Element of the carrier of multMagma(# (MGFC),S #)
(p1 * ps) * qi is Element of the carrier of multMagma(# (MGFC),S #)
the multF of multMagma(# (MGFC),S #) . ((p1 * ps),qi) is Element of the carrier of multMagma(# (MGFC),S #)
[(p1 * ps),qi] is non empty set
{(p1 * ps),qi} is non empty finite set
{(p1 * ps)} is non empty trivial finite 1 -element set
{{(p1 * ps),qi},{(p1 * ps)}} is non empty finite V44() set
the multF of multMagma(# (MGFC),S #) . [(p1 * ps),qi] is set
ps * qi is Element of the carrier of multMagma(# (MGFC),S #)
the multF of multMagma(# (MGFC),S #) . (ps,qi) is Element of the carrier of multMagma(# (MGFC),S #)
[ps,qi] is non empty set
{ps,qi} is non empty finite set
{ps} is non empty trivial finite 1 -element set
{{ps,qi},{ps}} is non empty finite V44() set
the multF of multMagma(# (MGFC),S #) . [ps,qi] is set
p1 * (ps * qi) is Element of the carrier of multMagma(# (MGFC),S #)
the multF of multMagma(# (MGFC),S #) . (p1,(ps * qi)) is Element of the carrier of multMagma(# (MGFC),S #)
[p1,(ps * qi)] is non empty set
{p1,(ps * qi)} is non empty finite set
{{p1,(ps * qi)},{p1}} is non empty finite V44() set
the multF of multMagma(# (MGFC),S #) . [p1,(ps * qi)] is set
[p1,ps] is non empty Element of [: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):]
( the multF of F_Complex || (MGFC)) . [p1,ps] is set
rng ( the multF of F_Complex || (MGFC)) is set
[(( the multF of F_Complex || (MGFC)) . [p1,ps]),qi] is non empty set
{(( the multF of F_Complex || (MGFC)) . [p1,ps]),qi} is non empty finite set
{(( the multF of F_Complex || (MGFC)) . [p1,ps])} is non empty trivial finite 1 -element set
{{(( the multF of F_Complex || (MGFC)) . [p1,ps]),qi},{(( the multF of F_Complex || (MGFC)) . [p1,ps])}} is non empty finite V44() set
[ps,qi] is non empty Element of [: the carrier of multMagma(# (MGFC),S #), the carrier of multMagma(# (MGFC),S #):]
( the multF of F_Complex || (MGFC)) . [ps,qi] is set
[p1,(( the multF of F_Complex || (MGFC)) . [ps,qi])] is non empty set
{p1,(( the multF of F_Complex || (MGFC)) . [ps,qi])} is non empty finite set
{{p1,(( the multF of F_Complex || (MGFC)) . [ps,qi])},{p1}} is non empty finite V44() set
multcomplex . [ps,qi] is V31() set
( the multF of F_Complex || (MGFC)) . [p1,(( the multF of F_Complex || (MGFC)) . [ps,qi])] is set
multcomplex . (ps,qi) is set
multcomplex . [ps,qi] is V31() set
multcomplex . (p1,(multcomplex . (ps,qi))) is set
[p1,(multcomplex . (ps,qi))] is non empty set
{p1,(multcomplex . (ps,qi))} is non empty finite set
{{p1,(multcomplex . (ps,qi))},{p1}} is non empty finite V44() set
multcomplex . [p1,(multcomplex . (ps,qi))] is V31() set
multcomplex . [p1,ps] is V31() set
( the multF of F_Complex || (MGFC)) . [(( the multF of F_Complex || (MGFC)) . [p1,ps]),qi] is set
multcomplex . (p1,ps) is set
multcomplex . [p1,ps] is V31() set
multcomplex . ((multcomplex . (p1,ps)),qi) is set
[(multcomplex . (p1,ps)),qi] is non empty set
{(multcomplex . (p1,ps)),qi} is non empty finite set
{(multcomplex . (p1,ps))} is non empty trivial finite 1 -element set
{{(multcomplex . (p1,ps)),qi},{(multcomplex . (p1,ps))}} is non empty finite V44() set
multcomplex . [(multcomplex . (p1,ps)),qi] is V31() set
cMGFC is non empty strict unital Group-like associative multMagma
the carrier of cMGFC is non empty set
the multF of cMGFC is Relation-like [: the carrier of cMGFC, the carrier of cMGFC:] -defined the carrier of cMGFC -valued Function-like V25([: the carrier of cMGFC, the carrier of cMGFC:], the carrier of cMGFC) associative V51( the carrier of cMGFC) Element of bool [:[: the carrier of cMGFC, the carrier of cMGFC:], the carrier of cMGFC:]
[: the carrier of cMGFC, the carrier of cMGFC:] is Relation-like set
[:[: the carrier of cMGFC, the carrier of cMGFC:], the carrier of cMGFC:] is Relation-like set
bool [:[: the carrier of cMGFC, the carrier of cMGFC:], the carrier of cMGFC:] is set
n is non empty strict unital Group-like associative multMagma
the carrier of n is non empty set
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like V25([: the carrier of n, the carrier of n:], the carrier of n) associative V51( the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
S is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(S) is non empty strict unital Group-like associative multMagma
the carrier of (S) is non empty set
(S) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of S, 1_ F_Complex } is set
the multF of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined the carrier of (S) -valued Function-like V25([: the carrier of (S), the carrier of (S):], the carrier of (S)) associative V51( the carrier of (S)) Element of bool [:[: the carrier of (S), the carrier of (S):], the carrier of (S):]
[: the carrier of (S), the carrier of (S):] is Relation-like set
[:[: the carrier of (S), the carrier of (S):], the carrier of (S):] is Relation-like set
bool [:[: the carrier of (S), the carrier of (S):], the carrier of (S):] is set
the multF of F_Complex || (S) is Relation-like Function-like set
[:(S),(S):] is Relation-like finite set
the multF of F_Complex | [:(S),(S):] is Relation-like [:(S),(S):] -defined [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like finite finite-support set
the multF of (F_Complex) is Relation-like [: the carrier of (F_Complex), the carrier of (F_Complex):] -defined the carrier of (F_Complex) -valued Function-like V25([: the carrier of (F_Complex), the carrier of (F_Complex):], the carrier of (F_Complex)) associative V51( the carrier of (F_Complex)) Element of bool [:[: the carrier of (F_Complex), the carrier of (F_Complex):], the carrier of (F_Complex):]
[: the carrier of (F_Complex), the carrier of (F_Complex):] is Relation-like set
[:[: the carrier of (F_Complex), the carrier of (F_Complex):], the carrier of (F_Complex):] is Relation-like set
bool [:[: the carrier of (F_Complex), the carrier of (F_Complex):], the carrier of (F_Complex):] is set
the multF of F_Complex || the carrier of (F_Complex) is Relation-like Function-like set
the multF of F_Complex | [: the carrier of (F_Complex), the carrier of (F_Complex):] is Relation-like [: the carrier of (F_Complex), the carrier of (F_Complex):] -defined [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like set
the multF of (F_Complex) || the carrier of (S) is Relation-like Function-like set
the multF of (F_Complex) | [: the carrier of (S), the carrier of (S):] is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (F_Complex), the carrier of (F_Complex):] -defined the carrier of (F_Complex) -valued Function-like set
cMGFC is non empty left_unital doubleLoopStr
the carrier of cMGFC is non empty set
0_. cMGFC is Relation-like NAT -defined the carrier of cMGFC -valued Function-like V25( NAT , the carrier of cMGFC) finite-Support Element of bool [:NAT, the carrier of cMGFC:]
[:NAT, the carrier of cMGFC:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of cMGFC:] is non empty non trivial non finite set
0. cMGFC is zero Element of the carrier of cMGFC
the ZeroF of cMGFC is Element of the carrier of cMGFC
K173( the carrier of cMGFC,NAT,(0. cMGFC)) is Relation-like NAT -defined the carrier of cMGFC -valued T-Sequence-like Function-like V25( NAT , the carrier of cMGFC) Element of bool [:NAT, the carrier of cMGFC:]
1_ cMGFC is Element of the carrier of cMGFC
- (1_ cMGFC) is Element of the carrier of cMGFC
(0_. cMGFC) +* (0,(- (1_ cMGFC))) is Relation-like NAT -defined the carrier of cMGFC -valued Function-like V25( NAT , the carrier of cMGFC) Element of bool [:NAT, the carrier of cMGFC:]
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal set
((0_. cMGFC) +* (0,(- (1_ cMGFC)))) +* (MGFC,(1_ cMGFC)) is Relation-like NAT -defined the carrier of cMGFC -valued Function-like V25( NAT , the carrier of cMGFC) Element of bool [:NAT, the carrier of cMGFC:]
fs is ordinal natural V31() real ext-real non negative integer finite cardinal set
(((0_. cMGFC) +* (0,(- (1_ cMGFC)))) +* (MGFC,(1_ cMGFC))) . fs is set
((0_. cMGFC) +* (0,(- (1_ cMGFC)))) . fs is set
(0_. cMGFC) . fs is set
MGFC + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is ordinal natural V31() real ext-real non negative integer finite cardinal set
(((0_. cMGFC) +* (0,(- (1_ cMGFC)))) +* (MGFC,(1_ cMGFC))) . S is set
MGFC + 0 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(1,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
[:NAT, the carrier of F_Complex:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of F_Complex:] is non empty non trivial non finite set
0_. F_Complex is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support with_roots Element of bool [:NAT, the carrier of F_Complex:]
K173( the carrier of F_Complex,NAT,(0. F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued T-Sequence-like Function-like V25( NAT , the carrier of F_Complex) V57() Element of bool [:NAT, the carrier of F_Complex:]
- (1_ F_Complex) is V31() right_complementable Element of the carrier of F_Complex
(0_. F_Complex) +* (0,(- (1_ F_Complex))) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (1,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
<%(- (1_ F_Complex)),(1_ F_Complex)%> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
MGFC is non empty left_unital doubleLoopStr
the carrier of MGFC is non empty set
1_ MGFC is Element of the carrier of MGFC
- (1_ MGFC) is Element of the carrier of MGFC
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC,MGFC) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) finite-Support Element of bool [:NAT, the carrier of MGFC:]
[:NAT, the carrier of MGFC:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of MGFC:] is non empty non trivial non finite set
0_. MGFC is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) finite-Support Element of bool [:NAT, the carrier of MGFC:]
0. MGFC is zero Element of the carrier of MGFC
the ZeroF of MGFC is Element of the carrier of MGFC
K173( the carrier of MGFC,NAT,(0. MGFC)) is Relation-like NAT -defined the carrier of MGFC -valued T-Sequence-like Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
(0_. MGFC) +* (0,(- (1_ MGFC))) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
((0_. MGFC) +* (0,(- (1_ MGFC)))) +* (cMGFC,(1_ MGFC)) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
(cMGFC,MGFC) . 0 is Element of the carrier of MGFC
(cMGFC,MGFC) . cMGFC is Element of the carrier of MGFC
(0_. MGFC) +* (cMGFC,(1_ MGFC)) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
((0_. MGFC) +* (cMGFC,(1_ MGFC))) +* (0,(- (1_ MGFC))) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
dom ((0_. MGFC) +* (cMGFC,(1_ MGFC))) is set
dom ((0_. MGFC) +* (0,(- (1_ MGFC)))) is set
MGFC is non empty left_unital doubleLoopStr
0. MGFC is zero Element of the carrier of MGFC
the carrier of MGFC is non empty set
the ZeroF of MGFC is Element of the carrier of MGFC
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal set
(cMGFC,MGFC) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) finite-Support Element of bool [:NAT, the carrier of MGFC:]
[:NAT, the carrier of MGFC:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of MGFC:] is non empty non trivial non finite set
0_. MGFC is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) finite-Support Element of bool [:NAT, the carrier of MGFC:]
K173( the carrier of MGFC,NAT,(0. MGFC)) is Relation-like NAT -defined the carrier of MGFC -valued T-Sequence-like Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
1_ MGFC is Element of the carrier of MGFC
- (1_ MGFC) is Element of the carrier of MGFC
(0_. MGFC) +* (0,(- (1_ MGFC))) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
((0_. MGFC) +* (0,(- (1_ MGFC)))) +* (cMGFC,(1_ MGFC)) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
n is ordinal natural V31() real ext-real non negative integer finite cardinal set
(cMGFC,MGFC) . n is set
(((0_. MGFC) +* (0,(- (1_ MGFC)))) +* (cMGFC,(1_ MGFC))) . n is set
((0_. MGFC) +* (0,(- (1_ MGFC)))) . n is set
(0_. MGFC) . n is set
MGFC is non empty non degenerated non trivial unital right_unital well-unital left_unital doubleLoopStr
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC,MGFC) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) finite-Support Element of bool [:NAT, the carrier of MGFC:]
the carrier of MGFC is non empty non trivial set
[:NAT, the carrier of MGFC:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of MGFC:] is non empty non trivial non finite set
0_. MGFC is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) finite-Support with_roots Element of bool [:NAT, the carrier of MGFC:]
0. MGFC is zero Element of the carrier of MGFC
the ZeroF of MGFC is Element of the carrier of MGFC
K173( the carrier of MGFC,NAT,(0. MGFC)) is Relation-like NAT -defined the carrier of MGFC -valued T-Sequence-like Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
1_ MGFC is Element of the carrier of MGFC
1. MGFC is non zero Element of the carrier of MGFC
the OneF of MGFC is Element of the carrier of MGFC
- (1_ MGFC) is Element of the carrier of MGFC
(0_. MGFC) +* (0,(- (1_ MGFC))) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
((0_. MGFC) +* (0,(- (1_ MGFC)))) +* (cMGFC,(1_ MGFC)) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
len (cMGFC,MGFC) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n is ordinal natural V31() real ext-real non negative integer finite cardinal set
(cMGFC,MGFC) . cMGFC is Element of the carrier of MGFC
n is ordinal natural V31() real ext-real non negative integer finite cardinal set
(cMGFC,MGFC) . n is set
cMGFC + 0 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC is non empty non degenerated non trivial unital right_unital well-unital left_unital doubleLoopStr
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC,MGFC) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) finite-Support Element of bool [:NAT, the carrier of MGFC:]
the carrier of MGFC is non empty non trivial set
[:NAT, the carrier of MGFC:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of MGFC:] is non empty non trivial non finite set
0_. MGFC is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) finite-Support with_roots Element of bool [:NAT, the carrier of MGFC:]
0. MGFC is zero Element of the carrier of MGFC
the ZeroF of MGFC is Element of the carrier of MGFC
K173( the carrier of MGFC,NAT,(0. MGFC)) is Relation-like NAT -defined the carrier of MGFC -valued T-Sequence-like Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
1_ MGFC is Element of the carrier of MGFC
1. MGFC is non zero Element of the carrier of MGFC
the OneF of MGFC is Element of the carrier of MGFC
- (1_ MGFC) is Element of the carrier of MGFC
(0_. MGFC) +* (0,(- (1_ MGFC))) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
((0_. MGFC) +* (0,(- (1_ MGFC)))) +* (cMGFC,(1_ MGFC)) is Relation-like NAT -defined the carrier of MGFC -valued Function-like V25( NAT , the carrier of MGFC) Element of bool [:NAT, the carrier of MGFC:]
len (cMGFC,MGFC) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
1 - 1 is V31() real ext-real integer rational Element of INT
1 -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (cMGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
n is V31() right_complementable Element of the carrier of F_Complex
eval ((cMGFC,F_Complex),n) is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (n,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
[n,cMGFC] is non empty set
{n,cMGFC} is non empty finite V67() set
{n} is non empty trivial finite 1 -element V67() set
{{n,cMGFC},{n}} is non empty finite V44() set
(power F_Complex) . [n,cMGFC] is set
((power F_Complex) . (n,cMGFC)) - 1 is V31() Element of COMPLEX
len (cMGFC,F_Complex) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
Sum fs is V31() right_complementable Element of the carrier of F_Complex
len fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
cMGFC + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len fs) - 1 is V31() real ext-real integer rational Element of INT
(len fs) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
((len fs) - 1) + 1 is V31() real ext-real integer rational Element of INT
((len fs) -' 1) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC,F_Complex) . 0 is V31() right_complementable Element of the carrier of F_Complex
((len fs) -' 1) |-> (0. F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite (len fs) -' 1 -element FinSequence-like FinSubsequence-like finite-support Element of ((len fs) -' 1) -tuples_on the carrier of F_Complex
((len fs) -' 1) -tuples_on the carrier of F_Complex is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
the carrier of F_Complex * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex * : len b₁ = (len fs) -' 1 } is set
Seg ((len fs) -' 1) is finite (len fs) -' 1 -element V67() V68() V69() V70() V71() V72() Element of bool NAT
K172((Seg ((len fs) -' 1)),(0. F_Complex)) is Relation-like Seg ((len fs) -' 1) -defined {(0. F_Complex)} -valued Function-like V25( Seg ((len fs) -' 1),{(0. F_Complex)}) finite FinSequence-like FinSubsequence-like V57() finite-support Element of bool [:(Seg ((len fs) -' 1)),{(0. F_Complex)}:]
{(0. F_Complex)} is non empty trivial finite 1 -element V67() set
[:(Seg ((len fs) -' 1)),{(0. F_Complex)}:] is Relation-like finite V57() set
bool [:(Seg ((len fs) -' 1)),{(0. F_Complex)}:] is finite V44() set
len (((len fs) -' 1) |-> (0. F_Complex)) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
<*((power F_Complex) . (n,cMGFC))*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex *
(((len fs) -' 1) |-> (0. F_Complex)) ^ <*((power F_Complex) . (n,cMGFC))*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like non empty finite K231(((len fs) -' 1),1) -element FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
K231(((len fs) -' 1),1) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
<*(- (1_ F_Complex))*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex *
<*(- (1_ F_Complex))*> ^ (((len fs) -' 1) |-> (0. F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like non empty finite K231(1,((len fs) -' 1)) -element FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
K231(1,((len fs) -' 1)) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg (len fs) is finite len fs -element V67() V68() V69() V70() V71() V72() Element of bool NAT
(len fs) -tuples_on the carrier of F_Complex is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
{ b₁ where b₁ is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex * : len b₁ = len fs } is set
qi is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite len fs -element FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
x is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite len fs -element FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
dom (((len fs) -' 1) |-> (0. F_Complex)) is finite (len fs) -' 1 -element V67() V68() V69() V70() V71() V72() Element of bool NAT
mc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(((len fs) -' 1) |-> (0. F_Complex)) . mc is set
i is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite len fs -element FinSequence-like FinSubsequence-like finite-support Element of (len fs) -tuples_on the carrier of F_Complex
dom i is finite len fs -element V67() V68() V69() V70() V71() V72() Element of bool NAT
mc is ordinal natural V31() real ext-real non negative integer finite cardinal set
i . mc is set
dom <*(- (1_ F_Complex))*> is non empty trivial finite 1 -element V67() V68() V69() V70() V71() V72() Element of bool NAT
len <*(- (1_ F_Complex))*> is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
jcf is ordinal natural V31() real ext-real non negative integer finite cardinal set
(len <*(- (1_ F_Complex))*>) + jcf is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(((len fs) -' 1) |-> (0. F_Complex)) . jcf is set
len ((((len fs) -' 1) |-> (0. F_Complex)) ^ <*((power F_Complex) . (n,cMGFC))*>) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
len <*((power F_Complex) . (n,cMGFC))*> is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len (((len fs) -' 1) |-> (0. F_Complex))) + (len <*((power F_Complex) . (n,cMGFC))*>) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
lc is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite len fs -element FinSequence-like FinSubsequence-like finite-support Element of (len fs) -tuples_on the carrier of F_Complex
dom lc is finite len fs -element V67() V68() V69() V70() V71() V72() Element of bool NAT
mc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
lc . mc is set
(((len fs) -' 1) |-> (0. F_Complex)) . mc is set
len i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
len lc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
i . 1 is set
i + lc is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite len fs -element FinSequence-like FinSubsequence-like finite-support Element of (len fs) -tuples_on the carrier of F_Complex
len (i + lc) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom (i + lc) is finite len fs -element V67() V68() V69() V70() V71() V72() Element of bool NAT
lc . (len fs) is set
mc is ordinal natural V31() real ext-real non negative integer finite cardinal set
(i + lc) . mc is set
fs . mc is set
(- (1_ F_Complex)) * (1_ F_Complex) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((- (1_ F_Complex)),(1_ F_Complex)) is V31() right_complementable Element of the carrier of F_Complex
[(- (1_ F_Complex)),(1_ F_Complex)] is non empty set
{(- (1_ F_Complex)),(1_ F_Complex)} is non empty finite V67() set
{(- (1_ F_Complex))} is non empty trivial finite 1 -element V67() set
{{(- (1_ F_Complex)),(1_ F_Complex)},{(- (1_ F_Complex))}} is non empty finite V44() set
the multF of F_Complex . [(- (1_ F_Complex)),(1_ F_Complex)] is set
(- (1_ F_Complex)) * (1_ F_Complex) is V31() Element of COMPLEX
lc . mc is set
(i + lc) . 1 is set
(- (1_ F_Complex)) + (0. F_Complex) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . ((- (1_ F_Complex)),(0. F_Complex)) is V31() right_complementable Element of the carrier of F_Complex
[(- (1_ F_Complex)),(0. F_Complex)] is non empty set
{(- (1_ F_Complex)),(0. F_Complex)} is non empty finite V67() set
{{(- (1_ F_Complex)),(0. F_Complex)},{(- (1_ F_Complex))}} is non empty finite V44() set
the addF of F_Complex . [(- (1_ F_Complex)),(0. F_Complex)] is set
(- (1_ F_Complex)) + (0. F_Complex) is V31() Element of COMPLEX
fs . 1 is set
(power F_Complex) . (n,0) is V31() right_complementable Element of the carrier of F_Complex
[n,0] is non empty set
{n,0} is non empty finite V67() set
{{n,0},{n}} is non empty finite V44() set
(power F_Complex) . [n,0] is set
((cMGFC,F_Complex) . 0) * ((power F_Complex) . (n,0)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . (((cMGFC,F_Complex) . 0),((power F_Complex) . (n,0))) is V31() right_complementable Element of the carrier of F_Complex
[((cMGFC,F_Complex) . 0),((power F_Complex) . (n,0))] is non empty set
{((cMGFC,F_Complex) . 0),((power F_Complex) . (n,0))} is non empty finite V67() set
{((cMGFC,F_Complex) . 0)} is non empty trivial finite 1 -element V67() set
{{((cMGFC,F_Complex) . 0),((power F_Complex) . (n,0))},{((cMGFC,F_Complex) . 0)}} is non empty finite V44() set
the multF of F_Complex . [((cMGFC,F_Complex) . 0),((power F_Complex) . (n,0))] is set
((cMGFC,F_Complex) . 0) * ((power F_Complex) . (n,0)) is V31() Element of COMPLEX
fs . (len fs) is set
(cMGFC,F_Complex) . ((len fs) -' 1) is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (n,((len fs) -' 1)) is V31() right_complementable Element of the carrier of F_Complex
[n,((len fs) -' 1)] is non empty set
{n,((len fs) -' 1)} is non empty finite V67() set
{{n,((len fs) -' 1)},{n}} is non empty finite V44() set
(power F_Complex) . [n,((len fs) -' 1)] is set
((cMGFC,F_Complex) . ((len fs) -' 1)) * ((power F_Complex) . (n,((len fs) -' 1))) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . (((cMGFC,F_Complex) . ((len fs) -' 1)),((power F_Complex) . (n,((len fs) -' 1)))) is V31() right_complementable Element of the carrier of F_Complex
[((cMGFC,F_Complex) . ((len fs) -' 1)),((power F_Complex) . (n,((len fs) -' 1)))] is non empty set
{((cMGFC,F_Complex) . ((len fs) -' 1)),((power F_Complex) . (n,((len fs) -' 1)))} is non empty finite V67() set
{((cMGFC,F_Complex) . ((len fs) -' 1))} is non empty trivial finite 1 -element V67() set
{{((cMGFC,F_Complex) . ((len fs) -' 1)),((power F_Complex) . (n,((len fs) -' 1)))},{((cMGFC,F_Complex) . ((len fs) -' 1))}} is non empty finite V44() set
the multF of F_Complex . [((cMGFC,F_Complex) . ((len fs) -' 1)),((power F_Complex) . (n,((len fs) -' 1)))] is set
((cMGFC,F_Complex) . ((len fs) -' 1)) * ((power F_Complex) . (n,((len fs) -' 1))) is V31() Element of COMPLEX
(1_ F_Complex) * ((power F_Complex) . (n,cMGFC)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((1_ F_Complex),((power F_Complex) . (n,cMGFC))) is V31() right_complementable Element of the carrier of F_Complex
[(1_ F_Complex),((power F_Complex) . (n,cMGFC))] is non empty set
{(1_ F_Complex),((power F_Complex) . (n,cMGFC))} is non empty finite V67() set
{(1_ F_Complex)} is non empty trivial finite 1 -element V67() set
{{(1_ F_Complex),((power F_Complex) . (n,cMGFC))},{(1_ F_Complex)}} is non empty finite V44() set
the multF of F_Complex . [(1_ F_Complex),((power F_Complex) . (n,cMGFC))] is set
(1_ F_Complex) * ((power F_Complex) . (n,cMGFC)) is V31() Element of COMPLEX
i . (len fs) is set
(i + lc) . (len fs) is set
(0. F_Complex) + ((power F_Complex) . (n,cMGFC)) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . ((0. F_Complex),((power F_Complex) . (n,cMGFC))) is V31() right_complementable Element of the carrier of F_Complex
[(0. F_Complex),((power F_Complex) . (n,cMGFC))] is non empty set
{(0. F_Complex),((power F_Complex) . (n,cMGFC))} is non empty finite V67() set
{{(0. F_Complex),((power F_Complex) . (n,cMGFC))},{(0. F_Complex)}} is non empty finite V44() set
the addF of F_Complex . [(0. F_Complex),((power F_Complex) . (n,cMGFC))] is set
(0. F_Complex) + ((power F_Complex) . (n,cMGFC)) is V31() Element of COMPLEX
mc -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
mc - 1 is V31() real ext-real integer rational Element of INT
- 1 is non empty V31() real ext-real non positive negative integer rational Element of INT
mc + (- 1) is V31() real ext-real integer rational Element of INT
1 + (- 1) is V31() real ext-real integer rational Element of INT
(cMGFC,F_Complex) . (mc -' 1) is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (n,(mc -' 1)) is V31() right_complementable Element of the carrier of F_Complex
[n,(mc -' 1)] is non empty set
{n,(mc -' 1)} is non empty finite V67() set
{{n,(mc -' 1)},{n}} is non empty finite V44() set
(power F_Complex) . [n,(mc -' 1)] is set
(0. F_Complex) * ((power F_Complex) . (n,(mc -' 1))) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((0. F_Complex),((power F_Complex) . (n,(mc -' 1)))) is V31() right_complementable Element of the carrier of F_Complex
[(0. F_Complex),((power F_Complex) . (n,(mc -' 1)))] is non empty set
{(0. F_Complex),((power F_Complex) . (n,(mc -' 1)))} is non empty finite V67() set
{{(0. F_Complex),((power F_Complex) . (n,(mc -' 1)))},{(0. F_Complex)}} is non empty finite V44() set
the multF of F_Complex . [(0. F_Complex),((power F_Complex) . (n,(mc -' 1)))] is set
(0. F_Complex) * ((power F_Complex) . (n,(mc -' 1))) is V31() Element of COMPLEX
lc . mc is set
i . mc is set
(0. F_Complex) + (0. F_Complex) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . ((0. F_Complex),(0. F_Complex)) is V31() right_complementable Element of the carrier of F_Complex
[(0. F_Complex),(0. F_Complex)] is non empty set
{(0. F_Complex),(0. F_Complex)} is non empty finite V67() set
{{(0. F_Complex),(0. F_Complex)},{(0. F_Complex)}} is non empty finite V44() set
the addF of F_Complex . [(0. F_Complex),(0. F_Complex)] is set
(0. F_Complex) + (0. F_Complex) is V31() Element of COMPLEX
Sum (((len fs) -' 1) |-> (0. F_Complex)) is V31() right_complementable Element of the carrier of F_Complex
Sum i is V31() right_complementable Element of the carrier of F_Complex
(- (1_ F_Complex)) + (0. F_Complex) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . ((- (1_ F_Complex)),(0. F_Complex)) is V31() right_complementable Element of the carrier of F_Complex
[(- (1_ F_Complex)),(0. F_Complex)] is non empty set
{(- (1_ F_Complex)),(0. F_Complex)} is non empty finite V67() set
{(- (1_ F_Complex))} is non empty trivial finite 1 -element V67() set
{{(- (1_ F_Complex)),(0. F_Complex)},{(- (1_ F_Complex))}} is non empty finite V44() set
the addF of F_Complex . [(- (1_ F_Complex)),(0. F_Complex)] is set
(- (1_ F_Complex)) + (0. F_Complex) is V31() Element of COMPLEX
Sum lc is V31() right_complementable Element of the carrier of F_Complex
(0. F_Complex) + ((power F_Complex) . (n,cMGFC)) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex . ((0. F_Complex),((power F_Complex) . (n,cMGFC))) is V31() right_complementable Element of the carrier of F_Complex
[(0. F_Complex),((power F_Complex) . (n,cMGFC))] is non empty set
{(0. F_Complex),((power F_Complex) . (n,cMGFC))} is non empty finite V67() set
{{(0. F_Complex),((power F_Complex) . (n,cMGFC))},{(0. F_Complex)}} is non empty finite V44() set
the addF of F_Complex . [(0. F_Complex),((power F_Complex) . (n,cMGFC))] is set
(0. F_Complex) + ((power F_Complex) . (n,cMGFC)) is V31() Element of COMPLEX
MGFC is V31() Element of COMPLEX
- MGFC is V31() Element of COMPLEX
(- (1_ F_Complex)) + ((power F_Complex) . (n,cMGFC)) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex . ((- (1_ F_Complex)),((power F_Complex) . (n,cMGFC))) is V31() right_complementable Element of the carrier of F_Complex
[(- (1_ F_Complex)),((power F_Complex) . (n,cMGFC))] is non empty set
{(- (1_ F_Complex)),((power F_Complex) . (n,cMGFC))} is non empty finite V67() set
{{(- (1_ F_Complex)),((power F_Complex) . (n,cMGFC))},{(- (1_ F_Complex))}} is non empty finite V44() set
the addF of F_Complex . [(- (1_ F_Complex)),((power F_Complex) . (n,cMGFC))] is set
(- (1_ F_Complex)) + ((power F_Complex) . (n,cMGFC)) is V31() Element of COMPLEX
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (MGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
Roots (MGFC,F_Complex) is finite Element of bool the carrier of F_Complex
(MGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
n is set
S is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC,F_Complex),S) is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (S,MGFC) is V31() right_complementable Element of the carrier of F_Complex
[S,MGFC] is non empty set
{S,MGFC} is non empty finite V67() set
{S} is non empty trivial finite 1 -element V67() set
{{S,MGFC},{S}} is non empty finite V44() set
(power F_Complex) . [S,MGFC] is set
((power F_Complex) . (S,MGFC)) - 1 is V31() Element of COMPLEX
S is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (S,MGFC) is V31() right_complementable Element of the carrier of F_Complex
[S,MGFC] is non empty set
{S,MGFC} is non empty finite V67() set
{S} is non empty trivial finite 1 -element V67() set
{{S,MGFC},{S}} is non empty finite V44() set
(power F_Complex) . [S,MGFC] is set
((power F_Complex) . (S,MGFC)) - 1 is V31() Element of COMPLEX
eval ((MGFC,F_Complex),S) is V31() right_complementable Element of the carrier of F_Complex
MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (cMGFC,MGFC) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,MGFC] is non empty set
{cMGFC,MGFC} is non empty finite V67() set
{cMGFC} is non empty trivial finite 1 -element V67() set
{{cMGFC,MGFC},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,MGFC] is set
n is V31() real ext-real Element of REAL
n |^ MGFC is V31() real ext-real Element of REAL
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n |^ MGFC is V31() real ext-real Element of REAL
n is V31() real ext-real Element of REAL
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power F_Complex) . (cMGFC,S) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,S] is non empty set
{cMGFC,S} is non empty finite V67() set
{{cMGFC,S},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,S] is set
n |^ S is V31() real ext-real Element of REAL
S + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(power F_Complex) . (cMGFC,(S + 1)) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,(S + 1)] is non empty set
{cMGFC,(S + 1)} is non empty finite V67() set
{{cMGFC,(S + 1)},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,(S + 1)] is set
n |^ (S + 1) is V31() real ext-real Element of REAL
((power F_Complex) . (cMGFC,S)) * cMGFC is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . (((power F_Complex) . (cMGFC,S)),cMGFC) is V31() right_complementable Element of the carrier of F_Complex
[((power F_Complex) . (cMGFC,S)),cMGFC] is non empty set
{((power F_Complex) . (cMGFC,S)),cMGFC} is non empty finite V67() set
{((power F_Complex) . (cMGFC,S))} is non empty trivial finite 1 -element V67() set
{{((power F_Complex) . (cMGFC,S)),cMGFC},{((power F_Complex) . (cMGFC,S))}} is non empty finite V44() set
the multF of F_Complex . [((power F_Complex) . (cMGFC,S)),cMGFC] is set
((power F_Complex) . (cMGFC,S)) * cMGFC is V31() Element of COMPLEX
n #Z S is V31() real ext-real Element of REAL
(n #Z S) * n is V31() real ext-real Element of REAL
n #Z 1 is V31() real ext-real Element of REAL
(n #Z S) * (n #Z 1) is V31() real ext-real Element of REAL
n #Z (S + 1) is V31() real ext-real Element of REAL
(power F_Complex) . (cMGFC,0) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,0] is non empty set
{cMGFC,0} is non empty finite V67() set
{{cMGFC,0},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,0] is set
n #Z 0 is V31() real ext-real Element of REAL
n |^ 0 is V31() real ext-real Element of REAL
n |^ MGFC is V31() real ext-real Element of REAL
n is V31() real ext-real Element of REAL
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (MGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
cMGFC is V31() real ext-real Element of REAL
cMGFC |^ MGFC is V31() real ext-real Element of REAL
(cMGFC |^ MGFC) - 1 is V31() real ext-real Element of REAL
n is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (n,MGFC) is V31() right_complementable Element of the carrier of F_Complex
[n,MGFC] is non empty set
{n,MGFC} is non empty finite V67() set
{n} is non empty trivial finite 1 -element V67() set
{{n,MGFC},{n}} is non empty finite V44() set
(power F_Complex) . [n,MGFC] is set
eval ((MGFC,F_Complex),n) is V31() right_complementable Element of the carrier of F_Complex
S is V31() real ext-real Element of REAL
S |^ MGFC is V31() real ext-real Element of REAL
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (MGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
BRoots (MGFC,F_Complex) is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support set
(MGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
((MGFC),1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
degree (BRoots (MGFC,F_Complex)) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
len (MGFC,F_Complex) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len (MGFC,F_Complex)) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC + 1) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
card (MGFC) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Roots (MGFC,F_Complex) is finite Element of bool the carrier of F_Complex
support (BRoots (MGFC,F_Complex)) is finite set
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (MGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
(MGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of MGFC, 1_ F_Complex } is set
((MGFC),1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
poly_with_roots (((MGFC),1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
len (MGFC,F_Complex) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len (MGFC,F_Complex)) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC,F_Complex) . ((len (MGFC,F_Complex)) -' 1) is V31() right_complementable Element of the carrier of F_Complex
(MGFC,F_Complex) . MGFC is V31() right_complementable Element of the carrier of F_Complex
BRoots (MGFC,F_Complex) is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support set
poly_with_roots (BRoots (MGFC,F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (MGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
cMGFC is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC,F_Complex),cMGFC) is V31() right_complementable Element of the carrier of F_Complex
n is V31() real ext-real integer set
n |^ MGFC is V31() real ext-real integer set
(n |^ MGFC) - 1 is V31() real ext-real integer rational Element of INT
S is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC,F_Complex),S) is V31() right_complementable Element of the carrier of F_Complex
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal set
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ = MGFC } is set
n is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : S₁[b₁] } is set
(2 * PI) * 1 is V31() real ext-real Element of REAL
((2 * PI) * 1) / n is V31() real ext-real Element of REAL
cos (((2 * PI) * 1) / n) is V31() real ext-real Element of REAL
sin (((2 * PI) * 1) / n) is V31() real ext-real Element of REAL
[**(cos (((2 * PI) * 1) / n)),(sin (((2 * PI) * 1) / n))**] is V31() right_complementable Element of the carrier of F_Complex
K104((sin (((2 * PI) * 1) / n)),<i>) is V31() set
K103((cos (((2 * PI) * 1) / n)),K104((sin (((2 * PI) * 1) / n)),<i>)) is V31() set
0 * <i> is V31() Element of COMPLEX
0 + (0 * <i>) is V31() Element of COMPLEX
(sin (((2 * PI) * 1) / n)) * <i> is V31() Element of COMPLEX
(cos (((2 * PI) * 1) / n)) + ((sin (((2 * PI) * 1) / n)) * <i>) is V31() Element of COMPLEX
{(0. F_Complex)} is non empty trivial finite 1 -element V67() Element of bool the carrier of F_Complex
NonZero F_Complex is non empty Element of bool the carrier of F_Complex
[#] F_Complex is non empty non proper Element of bool the carrier of F_Complex
{(0. F_Complex)} is non empty trivial finite 1 -element V67() set
([#] F_Complex) \ {(0. F_Complex)} is Element of bool the carrier of F_Complex
(n) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of n, 1_ F_Complex } is set
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ divides n } is set
p1 is set
ps is Element of the carrier of (F_Complex)
ord ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q is V31() real ext-real integer set
1 gcd q is ordinal natural V31() real ext-real non negative integer finite cardinal set
qc is Element of the carrier of (F_Complex)
ord qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n div 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 is non empty finite Element of bool the carrier of F_Complex
(p1,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
poly_with_roots ((p1,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
S is non empty finite Element of bool the carrier of F_Complex
cMGFC is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(S,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
poly_with_roots ((S,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
fs is non empty finite Element of bool the carrier of F_Complex
n is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(fs,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((fs,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(1) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ = 1 } is set
cMGFC is non empty finite Element of bool the carrier of F_Complex
(cMGFC,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
poly_with_roots ((cMGFC,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(1) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of 1, 1_ F_Complex } is set
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ divides 1 } is set
n is set
S is Element of the carrier of (F_Complex)
ord S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is Element of the carrier of (F_Complex)
ord S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Polynom-Ring F_Complex is non empty right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital V175() left_unital V185() V186() V187() domRing-like doubleLoopStr
the carrier of (Polynom-Ring F_Complex) is non empty set
<%(1_ F_Complex)%> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (cMGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
n is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
len n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom n is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Product n is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) is Relation-like [: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like V25([: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex)) associative Element of bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):]
[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] is Relation-like set
[:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is Relation-like set
bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is set
K228( the carrier of (Polynom-Ring F_Complex),n, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
(cMGFC) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of cMGFC, 1_ F_Complex } is set
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in (cMGFC) & ord b₁ <= a₁ ) } is set
fs is ordinal natural V31() real ext-real non negative integer finite cardinal set
Seg (len n) is finite len n -element V67() V68() V69() V70() V71() V72() Element of bool NAT
Seg fs is finite fs -element V67() V68() V69() V70() V71() V72() Element of bool NAT
n | (Seg fs) is Relation-like NAT -defined Seg fs -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
[:NAT, the carrier of (Polynom-Ring F_Complex):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring F_Complex):] is non empty non trivial non finite set
q is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product q is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),q, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
p1 is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
ps is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),ps, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
fs is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in (cMGFC) & ord b₁ <= q ) } is set
qc is set
p1 is Element of the carrier of (F_Complex)
ord p1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in (cMGFC) & ord b₁ <= cMGFC ) } is set
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ divides cMGFC } is set
qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in (cMGFC) & ord b₁ <= qc ) } is set
fs . qc is set
qc + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in (cMGFC) & ord b₁ <= qc + 1 ) } is set
fs . (qc + 1) is set
Seg qc is finite qc -element V67() V68() V69() V70() V71() V72() Element of bool NAT
n | (Seg qc) is Relation-like NAT -defined Seg qc -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
p1 is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product p1 is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),p1, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
Seg (qc + 1) is non empty finite qc + 1 -element K231(qc,1) -element V67() V68() V69() V70() V71() V72() Element of bool NAT
K231(qc,1) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n | (Seg (qc + 1)) is Relation-like NAT -defined Seg (qc + 1) -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
min ((qc + 1),(len n)) is ordinal natural V31() real ext-real non negative integer finite cardinal Element of REAL
ps is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
len ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qi is finite Element of bool the carrier of F_Complex
(qi,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
poly_with_roots ((qi,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
x is finite Element of bool the carrier of F_Complex
(x,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((x,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
x is finite Element of bool the carrier of F_Complex
(x,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((x,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(n | (Seg (qc + 1))) . (qc + 1) is set
n . (qc + 1) is set
ps . (qc + 1) is set
lc is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
ps | (Seg qc) is Relation-like NAT -defined Seg qc -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
<*lc*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring F_Complex) *
the carrier of (Polynom-Ring F_Complex) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring F_Complex)
p1 ^ <*lc*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),ps, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
(Product p1) * lc is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) . ((Product p1),lc) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
[(Product p1),lc] is non empty set
{(Product p1),lc} is non empty finite set
{(Product p1)} is non empty trivial finite 1 -element set
{{(Product p1),lc},{(Product p1)}} is non empty finite V44() set
the multF of (Polynom-Ring F_Complex) . [(Product p1),lc] is set
jcf is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(poly_with_roots ((x,1) -bag)) *' jcf is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
EmptyBag the carrier of F_Complex is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
mcf is set
scb is Element of the carrier of (F_Complex)
ord scb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
scb is Element of the carrier of (F_Complex)
ord scb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
poly_with_roots (EmptyBag the carrier of F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(poly_with_roots ((x,1) -bag)) *' (poly_with_roots (EmptyBag the carrier of F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
((x,1) -bag) + (EmptyBag the carrier of F_Complex) is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support set
poly_with_roots (((x,1) -bag) + (EmptyBag the carrier of F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ = qc + 1 } is set
((qc + 1)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
lcf is non empty finite Element of bool the carrier of F_Complex
(lcf,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((lcf,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
mcf is set
scb is Element of the carrier of (F_Complex)
ord scb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
x \/ lcf is non empty finite Element of bool the carrier of F_Complex
scb is Element of the carrier of (F_Complex)
ord scb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
scb is Element of the carrier of (F_Complex)
ord scb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
x /\ lcf is finite Element of bool the carrier of F_Complex
scb is set
cb is Element of the carrier of (F_Complex)
ord cb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
x is Element of the carrier of (F_Complex)
ord x is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(poly_with_roots ((x,1) -bag)) *' (poly_with_roots ((lcf,1) -bag)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
((x,1) -bag) + ((lcf,1) -bag) is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support set
poly_with_roots (((x,1) -bag) + ((lcf,1) -bag)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in (cMGFC) & ord b₁ <= 1 ) } is set
fs . 1 is set
qc is finite Element of bool the carrier of F_Complex
(qc,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
poly_with_roots ((qc,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
n | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
n . 1 is set
p1 is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
<*(n . 1)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
<*ps*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring F_Complex) *
the carrier of (Polynom-Ring F_Complex) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring F_Complex)
Product <*ps*> is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),<*ps*>, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ = 1 } is set
qi is non empty finite Element of bool the carrier of F_Complex
(qi,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((qi,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
i is set
x is Element of the carrier of (F_Complex)
ord x is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
x is Element of the carrier of (F_Complex)
ord x is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in (cMGFC) & ord b₁ <= len n ) } is set
fs . (len n) is set
((cMGFC),1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
n | (Seg (len n)) is Relation-like NAT -defined Seg (len n) -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
p1 is set
ps is Element of the carrier of (F_Complex)
ord ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q is finite Element of bool the carrier of F_Complex
ps is Element of the carrier of (F_Complex)
ord ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
poly_with_roots (((cMGFC),1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
p1 is finite Element of bool the carrier of F_Complex
(p1,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((p1,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg cMGFC is non empty finite cMGFC -element V67() V68() V69() V70() V71() V72() Element of bool NAT
(cMGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (cMGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
(cMGFC) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
n is ordinal natural V31() real ext-real non negative integer finite cardinal set
n + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
(fs) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
fs is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(fs) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
q is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
fs is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
dom S is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Seg n is finite n -element V67() V68() V69() V70() V71() V72() Element of bool NAT
S | (Seg n) is Relation-like NAT -defined Seg n -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
[:NAT, the carrier of (Polynom-Ring F_Complex):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring F_Complex):] is non empty non trivial non finite set
len S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S /. q is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
S . q is set
(q) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
qc is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(qc) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
q is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
<*q*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring F_Complex) *
the carrier of (Polynom-Ring F_Complex) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring F_Complex)
fs is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
<*<%(1_ F_Complex)%>*> is Relation-like NAT -defined bool [:NAT, the carrier of F_Complex:] -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of (bool [:NAT, the carrier of F_Complex:]) *
(bool [:NAT, the carrier of F_Complex:]) * is functional non empty FinSequence-membered FinSequenceSet of bool [:NAT, the carrier of F_Complex:]
fs ^ <*<%(1_ F_Complex)%>*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like finite-support set
qc is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product qc is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) is Relation-like [: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like V25([: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex)) associative Element of bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):]
[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] is Relation-like set
[:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is Relation-like set
bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is set
K228( the carrier of (Polynom-Ring F_Complex),qc, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
dom qc is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
p1 is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(cMGFC) *' p1 is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
len fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S | (Seg cMGFC) is Relation-like NAT -defined Seg cMGFC -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
S . cMGFC is set
len <*<%(1_ F_Complex)%>*> is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
i is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qc . i is set
(i) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
fs . i is set
S . i is set
dom <*<%(1_ F_Complex)%>*> is non empty trivial finite 1 -element V67() V68() V69() V70() V71() V72() Element of bool NAT
qc . cMGFC is set
<*<%(1_ F_Complex)%>*> . 1 is set
Product fs is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),fs, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
i is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
(Product fs) * i is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) . ((Product fs),i) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
[(Product fs),i] is non empty set
{(Product fs),i} is non empty finite set
{(Product fs)} is non empty trivial finite 1 -element set
{{(Product fs),i},{(Product fs)}} is non empty finite V44() set
the multF of (Polynom-Ring F_Complex) . [(Product fs),i] is set
x is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
x *' <%(1_ F_Complex)%> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
qi is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
<*(cMGFC)*> is Relation-like NAT -defined bool [:NAT, the carrier of F_Complex:] -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of (bool [:NAT, the carrier of F_Complex:]) *
fs ^ <*(cMGFC)*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like finite-support set
Product S is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),S, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
(Product fs) * ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) . ((Product fs),ps) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
[(Product fs),ps] is non empty set
{(Product fs),ps} is non empty finite set
{{(Product fs),ps},{(Product fs)}} is non empty finite V44() set
the multF of (Polynom-Ring F_Complex) . [(Product fs),ps] is set
- 1 is non empty V31() real ext-real non positive negative integer rational Element of INT
n is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
H₁(n) . S is V31() right_complementable Element of the carrier of F_Complex
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
H₁(n) . S is V31() right_complementable Element of the carrier of F_Complex
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
H₁(n) . S is V31() right_complementable Element of the carrier of F_Complex
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(n) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
H₁(n) . 0 is V31() right_complementable Element of the carrier of F_Complex
S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) . S is V31() right_complementable Element of the carrier of F_Complex
Seg n is non empty finite n -element V67() V68() V69() V70() V71() V72() Element of bool NAT
(n,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (n,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
(n) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
fs is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
S is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product S is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) is Relation-like [: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like V25([: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex)) associative Element of bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):]
[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] is Relation-like set
[:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is Relation-like set
bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is set
K228( the carrier of (Polynom-Ring F_Complex),S, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
dom S is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
(n) *' fs is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
len S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg (len S) is finite len S -element V67() V68() V69() V70() V71() V72() Element of bool NAT
q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg q is finite q -element V67() V68() V69() V70() V71() V72() Element of bool NAT
S | (Seg q) is Relation-like NAT -defined Seg q -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
[:NAT, the carrier of (Polynom-Ring F_Complex):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring F_Complex):] is non empty non trivial non finite set
q + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg (q + 1) is non empty finite q + 1 -element K231(q,1) -element V67() V68() V69() V70() V71() V72() Element of bool NAT
K231(q,1) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S | (Seg (q + 1)) is Relation-like NAT -defined Seg (q + 1) -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
S . (q + 1) is set
p1 is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product p1 is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),p1, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
<*> the carrier of (Polynom-Ring F_Complex) is Relation-like non-empty empty-yielding NAT -defined the carrier of (Polynom-Ring F_Complex) -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of the carrier of (Polynom-Ring F_Complex) *
the carrier of (Polynom-Ring F_Complex) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring F_Complex)
Seg 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal 0 -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of bool NAT
S | (Seg 0) is Relation-like non-empty empty-yielding NAT -defined Seg 0 -defined NAT -defined RAT -valued the carrier of (Polynom-Ring F_Complex) -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
<*(S . (q + 1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
(<*> the carrier of (Polynom-Ring F_Complex)) ^ <*(S . (q + 1))*> is Relation-like NAT -defined Function-like non empty finite K231({},1) -element FinSequence-like FinSubsequence-like finite-support set
K231({},1) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
ps is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
qc is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
ps . 0 is V31() right_complementable Element of the carrier of F_Complex
S . 1 is set
qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
ps . qi is V31() right_complementable Element of the carrier of F_Complex
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
ps is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
Product p1 is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),p1, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
qc is set
ps . 0 is V31() right_complementable Element of the carrier of F_Complex
qc is set
S . (q + 1) is set
ps is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),ps, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
qi is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
qi . 0 is V31() right_complementable Element of the carrier of F_Complex
p1 is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
lc is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
mc is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
Product lc is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),lc, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
mc . 0 is V31() right_complementable Element of the carrier of F_Complex
lc is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
mc is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
Product lc is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),lc, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
mc . 0 is V31() right_complementable Element of the carrier of F_Complex
<*p1*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring F_Complex) *
the carrier of (Polynom-Ring F_Complex) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring F_Complex)
lc ^ <*p1*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
(Product lc) * p1 is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) . ((Product lc),p1) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
[(Product lc),p1] is non empty set
{(Product lc),p1} is non empty finite set
{(Product lc)} is non empty trivial finite 1 -element set
{{(Product lc),p1},{(Product lc)}} is non empty finite V44() set
the multF of (Polynom-Ring F_Complex) . [(Product lc),p1] is set
i is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
mc *' i is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
jcf is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len jcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Sum jcf is V31() right_complementable Element of the carrier of F_Complex
dom jcf is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
jcf . 1 is set
lcf is V31() right_complementable Element of the carrier of F_Complex
<*lcf*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex *
the carrier of F_Complex * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
1 -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
mc . (1 -' 1) is V31() right_complementable Element of the carrier of F_Complex
(0 + 1) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
i . ((0 + 1) -' 1) is V31() right_complementable Element of the carrier of F_Complex
(mc . (1 -' 1)) * (i . ((0 + 1) -' 1)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((mc . (1 -' 1)),(i . ((0 + 1) -' 1))) is V31() right_complementable Element of the carrier of F_Complex
[(mc . (1 -' 1)),(i . ((0 + 1) -' 1))] is non empty set
{(mc . (1 -' 1)),(i . ((0 + 1) -' 1))} is non empty finite V67() set
{(mc . (1 -' 1))} is non empty trivial finite 1 -element V67() set
{{(mc . (1 -' 1)),(i . ((0 + 1) -' 1))},{(mc . (1 -' 1))}} is non empty finite V44() set
the multF of F_Complex . [(mc . (1 -' 1)),(i . ((0 + 1) -' 1))] is set
(mc . (1 -' 1)) * (i . ((0 + 1) -' 1)) is V31() Element of COMPLEX
mc . ((0 + 1) -' 1) is V31() right_complementable Element of the carrier of F_Complex
i . 0 is V31() right_complementable Element of the carrier of F_Complex
(mc . ((0 + 1) -' 1)) * (i . 0) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((mc . ((0 + 1) -' 1)),(i . 0)) is V31() right_complementable Element of the carrier of F_Complex
[(mc . ((0 + 1) -' 1)),(i . 0)] is non empty set
{(mc . ((0 + 1) -' 1)),(i . 0)} is non empty finite V67() set
{(mc . ((0 + 1) -' 1))} is non empty trivial finite 1 -element V67() set
{{(mc . ((0 + 1) -' 1)),(i . 0)},{(mc . ((0 + 1) -' 1))}} is non empty finite V44() set
the multF of F_Complex . [(mc . ((0 + 1) -' 1)),(i . 0)] is set
(mc . ((0 + 1) -' 1)) * (i . 0) is V31() Element of COMPLEX
(mc . 0) * (i . 0) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((mc . 0),(i . 0)) is V31() right_complementable Element of the carrier of F_Complex
[(mc . 0),(i . 0)] is non empty set
{(mc . 0),(i . 0)} is non empty finite V67() set
{(mc . 0)} is non empty trivial finite 1 -element V67() set
{{(mc . 0),(i . 0)},{(mc . 0)}} is non empty finite V44() set
the multF of F_Complex . [(mc . 0),(i . 0)] is set
(mc . 0) * (i . 0) is V31() Element of COMPLEX
(mc . 0) * (1_ F_Complex) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((mc . 0),(1_ F_Complex)) is V31() right_complementable Element of the carrier of F_Complex
[(mc . 0),(1_ F_Complex)] is non empty set
{(mc . 0),(1_ F_Complex)} is non empty finite V67() set
{{(mc . 0),(1_ F_Complex)},{(mc . 0)}} is non empty finite V44() set
the multF of F_Complex . [(mc . 0),(1_ F_Complex)] is set
(mc . 0) * (1_ F_Complex) is V31() Element of COMPLEX
mcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qi . mcf is V31() right_complementable Element of the carrier of F_Complex
mcf + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
scb is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len scb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Sum scb is V31() right_complementable Element of the carrier of F_Complex
dom scb is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
cb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
scb . cb is set
cb -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
mc . (cb -' 1) is V31() right_complementable Element of the carrier of F_Complex
(mcf + 1) -' cb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
i . ((mcf + 1) -' cb) is V31() right_complementable Element of the carrier of F_Complex
i . ((mcf + 1) -' cb) is V31() right_complementable Element of the carrier of F_Complex
i . ((mcf + 1) -' cb) is V31() right_complementable Element of the carrier of F_Complex
i . ((mcf + 1) -' cb) is V31() right_complementable Element of the carrier of F_Complex
(mc . (cb -' 1)) * (i . ((mcf + 1) -' cb)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((mc . (cb -' 1)),(i . ((mcf + 1) -' cb))) is V31() right_complementable Element of the carrier of F_Complex
[(mc . (cb -' 1)),(i . ((mcf + 1) -' cb))] is non empty set
{(mc . (cb -' 1)),(i . ((mcf + 1) -' cb))} is non empty finite V67() set
{(mc . (cb -' 1))} is non empty trivial finite 1 -element V67() set
{{(mc . (cb -' 1)),(i . ((mcf + 1) -' cb))},{(mc . (cb -' 1))}} is non empty finite V44() set
the multF of F_Complex . [(mc . (cb -' 1)),(i . ((mcf + 1) -' cb))] is set
(mc . (cb -' 1)) * (i . ((mcf + 1) -' cb)) is V31() Element of COMPLEX
x is V31() real ext-real integer set
pi1 is V31() real ext-real integer set
x * pi1 is V31() real ext-real integer rational Element of INT
jcf is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len jcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Sum jcf is V31() right_complementable Element of the carrier of F_Complex
dom jcf is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
jcf . 1 is set
lcf is V31() right_complementable Element of the carrier of F_Complex
<*lcf*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex *
the carrier of F_Complex * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
1 -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
mc . (1 -' 1) is V31() right_complementable Element of the carrier of F_Complex
(0 + 1) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
i . ((0 + 1) -' 1) is V31() right_complementable Element of the carrier of F_Complex
(mc . (1 -' 1)) * (i . ((0 + 1) -' 1)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((mc . (1 -' 1)),(i . ((0 + 1) -' 1))) is V31() right_complementable Element of the carrier of F_Complex
[(mc . (1 -' 1)),(i . ((0 + 1) -' 1))] is non empty set
{(mc . (1 -' 1)),(i . ((0 + 1) -' 1))} is non empty finite V67() set
{(mc . (1 -' 1))} is non empty trivial finite 1 -element V67() set
{{(mc . (1 -' 1)),(i . ((0 + 1) -' 1))},{(mc . (1 -' 1))}} is non empty finite V44() set
the multF of F_Complex . [(mc . (1 -' 1)),(i . ((0 + 1) -' 1))] is set
(mc . (1 -' 1)) * (i . ((0 + 1) -' 1)) is V31() Element of COMPLEX
mc . ((0 + 1) -' 1) is V31() right_complementable Element of the carrier of F_Complex
i . 0 is V31() right_complementable Element of the carrier of F_Complex
(mc . ((0 + 1) -' 1)) * (i . 0) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((mc . ((0 + 1) -' 1)),(i . 0)) is V31() right_complementable Element of the carrier of F_Complex
[(mc . ((0 + 1) -' 1)),(i . 0)] is non empty set
{(mc . ((0 + 1) -' 1)),(i . 0)} is non empty finite V67() set
{(mc . ((0 + 1) -' 1))} is non empty trivial finite 1 -element V67() set
{{(mc . ((0 + 1) -' 1)),(i . 0)},{(mc . ((0 + 1) -' 1))}} is non empty finite V44() set
the multF of F_Complex . [(mc . ((0 + 1) -' 1)),(i . 0)] is set
(mc . ((0 + 1) -' 1)) * (i . 0) is V31() Element of COMPLEX
(mc . 0) * (i . 0) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((mc . 0),(i . 0)) is V31() right_complementable Element of the carrier of F_Complex
[(mc . 0),(i . 0)] is non empty set
{(mc . 0),(i . 0)} is non empty finite V67() set
{(mc . 0)} is non empty trivial finite 1 -element V67() set
{{(mc . 0),(i . 0)},{(mc . 0)}} is non empty finite V44() set
the multF of F_Complex . [(mc . 0),(i . 0)] is set
(mc . 0) * (i . 0) is V31() Element of COMPLEX
((q + 1)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
mcf is V31() real ext-real integer set
x is V31() Element of COMPLEX
scb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qi . scb is V31() right_complementable Element of the carrier of F_Complex
scb + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cb is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len cb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Sum cb is V31() right_complementable Element of the carrier of F_Complex
dom cb is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
x is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cb . x is set
(scb + 1) -' x is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
i . ((scb + 1) -' x) is V31() right_complementable Element of the carrier of F_Complex
x -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
mc . (x -' 1) is V31() right_complementable Element of the carrier of F_Complex
(mc . (x -' 1)) * (i . ((scb + 1) -' x)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((mc . (x -' 1)),(i . ((scb + 1) -' x))) is V31() right_complementable Element of the carrier of F_Complex
[(mc . (x -' 1)),(i . ((scb + 1) -' x))] is non empty set
{(mc . (x -' 1)),(i . ((scb + 1) -' x))} is non empty finite V67() set
{(mc . (x -' 1))} is non empty trivial finite 1 -element V67() set
{{(mc . (x -' 1)),(i . ((scb + 1) -' x))},{(mc . (x -' 1))}} is non empty finite V44() set
the multF of F_Complex . [(mc . (x -' 1)),(i . ((scb + 1) -' x))] is set
(mc . (x -' 1)) * (i . ((scb + 1) -' x)) is V31() Element of COMPLEX
pi1 is V31() real ext-real integer set
fl1pj1i is V31() real ext-real integer set
pi1 * fl1pj1i is V31() real ext-real integer rational Element of INT
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(0 + 1) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal 0 -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of bool NAT
S | (Seg 0) is Relation-like non-empty empty-yielding NAT -defined Seg 0 -defined NAT -defined RAT -valued the carrier of (Polynom-Ring F_Complex) -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
[:NAT, the carrier of (Polynom-Ring F_Complex):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring F_Complex):] is non empty non trivial non finite set
q is ordinal natural V31() real ext-real non negative integer finite cardinal set
Seg q is finite q -element V67() V68() V69() V70() V71() V72() Element of bool NAT
S | (Seg q) is Relation-like NAT -defined Seg q -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
q is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom q is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
S | (Seg n) is Relation-like NAT -defined Seg n -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
q . n is set
qc is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
p1 is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
Product qc is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),qc, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
p1 . 0 is V31() right_complementable Element of the carrier of F_Complex
ps is ordinal natural V31() real ext-real non negative integer finite cardinal set
ps + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n,F_Complex) . ps is set
qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qi + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
i is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Sum i is V31() right_complementable Element of the carrier of F_Complex
dom i is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
x is V31() Element of COMPLEX
x is V31() Element of COMPLEX
x is V31() Element of COMPLEX
x is V31() Element of COMPLEX
x is V31() Element of COMPLEX
(1,1) -cut i is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
1 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
((1 + 1),(len i)) -cut i is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
((1,1) -cut i) ^ (((1 + 1),(len i)) -cut i) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
Sum (((1 + 1),(len i)) -cut i) is V31() right_complementable Element of the carrier of F_Complex
lcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom (((1 + 1),(len i)) -cut i) is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
(((1 + 1),(len i)) -cut i) . lcf is set
len (((1 + 1),(len i)) -cut i) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len (((1 + 1),(len i)) -cut i)) + (1 + 1) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len i) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(((1 + 1),(len i)) -cut i) . lcf is set
ps -' lcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
H₁(n) . (ps -' lcf) is V31() right_complementable Element of the carrier of F_Complex
fs . lcf is V31() right_complementable Element of the carrier of F_Complex
cb is ordinal natural V31() real ext-real non negative integer finite cardinal set
cb + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
lcf + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(len (((1 + 1),(len i)) -cut i)) + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
1 + lcf is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
i . (1 + lcf) is set
(1 + lcf) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs . ((1 + lcf) -' 1) is V31() right_complementable Element of the carrier of F_Complex
(ps + 1) -' (1 + lcf) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
H₁(n) . ((ps + 1) -' (1 + lcf)) is V31() right_complementable Element of the carrier of F_Complex
(fs . ((1 + lcf) -' 1)) * (H₁(n) . ((ps + 1) -' (1 + lcf))) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((fs . ((1 + lcf) -' 1)),(H₁(n) . ((ps + 1) -' (1 + lcf)))) is V31() right_complementable Element of the carrier of F_Complex
[(fs . ((1 + lcf) -' 1)),(H₁(n) . ((ps + 1) -' (1 + lcf)))] is non empty set
{(fs . ((1 + lcf) -' 1)),(H₁(n) . ((ps + 1) -' (1 + lcf)))} is non empty finite V67() set
{(fs . ((1 + lcf) -' 1))} is non empty trivial finite 1 -element V67() set
{{(fs . ((1 + lcf) -' 1)),(H₁(n) . ((ps + 1) -' (1 + lcf)))},{(fs . ((1 + lcf) -' 1))}} is non empty finite V44() set
the multF of F_Complex . [(fs . ((1 + lcf) -' 1)),(H₁(n) . ((ps + 1) -' (1 + lcf)))] is set
(fs . ((1 + lcf) -' 1)) * (H₁(n) . ((ps + 1) -' (1 + lcf))) is V31() Element of COMPLEX
(lcf + 1) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs . ((lcf + 1) -' 1) is V31() right_complementable Element of the carrier of F_Complex
(ps + 1) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
((ps + 1) -' 1) -' lcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
H₁(n) . (((ps + 1) -' 1) -' lcf) is V31() right_complementable Element of the carrier of F_Complex
(fs . ((lcf + 1) -' 1)) * (H₁(n) . (((ps + 1) -' 1) -' lcf)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((fs . ((lcf + 1) -' 1)),(H₁(n) . (((ps + 1) -' 1) -' lcf))) is V31() right_complementable Element of the carrier of F_Complex
[(fs . ((lcf + 1) -' 1)),(H₁(n) . (((ps + 1) -' 1) -' lcf))] is non empty set
{(fs . ((lcf + 1) -' 1)),(H₁(n) . (((ps + 1) -' 1) -' lcf))} is non empty finite V67() set
{(fs . ((lcf + 1) -' 1))} is non empty trivial finite 1 -element V67() set
{{(fs . ((lcf + 1) -' 1)),(H₁(n) . (((ps + 1) -' 1) -' lcf))},{(fs . ((lcf + 1) -' 1))}} is non empty finite V44() set
the multF of F_Complex . [(fs . ((lcf + 1) -' 1)),(H₁(n) . (((ps + 1) -' 1) -' lcf))] is set
(fs . ((lcf + 1) -' 1)) * (H₁(n) . (((ps + 1) -' 1) -' lcf)) is V31() Element of COMPLEX
(fs . lcf) * (H₁(n) . (((ps + 1) -' 1) -' lcf)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((fs . lcf),(H₁(n) . (((ps + 1) -' 1) -' lcf))) is V31() right_complementable Element of the carrier of F_Complex
[(fs . lcf),(H₁(n) . (((ps + 1) -' 1) -' lcf))] is non empty set
{(fs . lcf),(H₁(n) . (((ps + 1) -' 1) -' lcf))} is non empty finite V67() set
{(fs . lcf)} is non empty trivial finite 1 -element V67() set
{{(fs . lcf),(H₁(n) . (((ps + 1) -' 1) -' lcf))},{(fs . lcf)}} is non empty finite V44() set
the multF of F_Complex . [(fs . lcf),(H₁(n) . (((ps + 1) -' 1) -' lcf))] is set
(fs . lcf) * (H₁(n) . (((ps + 1) -' 1) -' lcf)) is V31() Element of COMPLEX
scb is V31() real ext-real integer set
mcf is V31() real ext-real integer set
scb * mcf is V31() real ext-real integer rational Element of INT
(((1 + 1),(len i)) -cut i) . lcf is set
(1 + 1) + cb is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
i . ((1 + 1) + cb) is set
jcf is V31() Element of COMPLEX
i . 1 is set
mcf is V31() right_complementable Element of the carrier of F_Complex
<*mcf*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex *
the carrier of F_Complex * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Complex
mcf + (Sum (((1 + 1),(len i)) -cut i)) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . (mcf,(Sum (((1 + 1),(len i)) -cut i))) is V31() right_complementable Element of the carrier of F_Complex
[mcf,(Sum (((1 + 1),(len i)) -cut i))] is non empty set
{mcf,(Sum (((1 + 1),(len i)) -cut i))} is non empty finite V67() set
{mcf} is non empty trivial finite 1 -element V67() set
{{mcf,(Sum (((1 + 1),(len i)) -cut i))},{mcf}} is non empty finite V44() set
the addF of F_Complex . [mcf,(Sum (((1 + 1),(len i)) -cut i))] is set
mcf + (Sum (((1 + 1),(len i)) -cut i)) is V31() Element of COMPLEX
scb is V31() Element of COMPLEX
lc is V31() real ext-real integer set
lcf is V31() real ext-real integer set
lc - lcf is V31() real ext-real integer rational Element of INT
cb is V31() real ext-real integer set
1 -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs . (1 -' 1) is V31() right_complementable Element of the carrier of F_Complex
(ps + 1) -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
H₁(n) . ((ps + 1) -' 1) is V31() right_complementable Element of the carrier of F_Complex
(fs . (1 -' 1)) * (H₁(n) . ((ps + 1) -' 1)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((fs . (1 -' 1)),(H₁(n) . ((ps + 1) -' 1))) is V31() right_complementable Element of the carrier of F_Complex
[(fs . (1 -' 1)),(H₁(n) . ((ps + 1) -' 1))] is non empty set
{(fs . (1 -' 1)),(H₁(n) . ((ps + 1) -' 1))} is non empty finite V67() set
{(fs . (1 -' 1))} is non empty trivial finite 1 -element V67() set
{{(fs . (1 -' 1)),(H₁(n) . ((ps + 1) -' 1))},{(fs . (1 -' 1))}} is non empty finite V44() set
the multF of F_Complex . [(fs . (1 -' 1)),(H₁(n) . ((ps + 1) -' 1))] is set
(fs . (1 -' 1)) * (H₁(n) . ((ps + 1) -' 1)) is V31() Element of COMPLEX
fs . 0 is V31() right_complementable Element of the carrier of F_Complex
H₁(n) . qi is V31() right_complementable Element of the carrier of F_Complex
(fs . 0) * (H₁(n) . qi) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((fs . 0),(H₁(n) . qi)) is V31() right_complementable Element of the carrier of F_Complex
[(fs . 0),(H₁(n) . qi)] is non empty set
{(fs . 0),(H₁(n) . qi)} is non empty finite V67() set
{(fs . 0)} is non empty trivial finite 1 -element V67() set
{{(fs . 0),(H₁(n) . qi)},{(fs . 0)}} is non empty finite V44() set
the multF of F_Complex . [(fs . 0),(H₁(n) . qi)] is set
(fs . 0) * (H₁(n) . qi) is V31() Element of COMPLEX
H₁(n) . ps is set
(- (1_ F_Complex)) * (H₁(n) . qi) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((- (1_ F_Complex)),(H₁(n) . qi)) is V31() right_complementable Element of the carrier of F_Complex
[(- (1_ F_Complex)),(H₁(n) . qi)] is non empty set
{(- (1_ F_Complex)),(H₁(n) . qi)} is non empty finite V67() set
{(- (1_ F_Complex))} is non empty trivial finite 1 -element V67() set
{{(- (1_ F_Complex)),(H₁(n) . qi)},{(- (1_ F_Complex))}} is non empty finite V44() set
the multF of F_Complex . [(- (1_ F_Complex)),(H₁(n) . qi)] is set
(- (1_ F_Complex)) * (H₁(n) . qi) is V31() Element of COMPLEX
(1_ F_Complex) * (H₁(n) . qi) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((1_ F_Complex),(H₁(n) . qi)) is V31() right_complementable Element of the carrier of F_Complex
[(1_ F_Complex),(H₁(n) . qi)] is non empty set
{(1_ F_Complex),(H₁(n) . qi)} is non empty finite V67() set
{(1_ F_Complex)} is non empty trivial finite 1 -element V67() set
{{(1_ F_Complex),(H₁(n) . qi)},{(1_ F_Complex)}} is non empty finite V44() set
the multF of F_Complex . [(1_ F_Complex),(H₁(n) . qi)] is set
(1_ F_Complex) * (H₁(n) . qi) is V31() Element of COMPLEX
- ((1_ F_Complex) * (H₁(n) . qi)) is V31() right_complementable Element of the carrier of F_Complex
- (H₁(n) . qi) is V31() right_complementable Element of the carrier of F_Complex
- mcf is V31() right_complementable Element of the carrier of F_Complex
(- (H₁(n) . qi)) + (- mcf) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex . ((- (H₁(n) . qi)),(- mcf)) is V31() right_complementable Element of the carrier of F_Complex
[(- (H₁(n) . qi)),(- mcf)] is non empty set
{(- (H₁(n) . qi)),(- mcf)} is non empty finite V67() set
{(- (H₁(n) . qi))} is non empty trivial finite 1 -element V67() set
{{(- (H₁(n) . qi)),(- mcf)},{(- (H₁(n) . qi))}} is non empty finite V44() set
the addF of F_Complex . [(- (H₁(n) . qi)),(- mcf)] is set
(- (H₁(n) . qi)) + (- mcf) is V31() Element of COMPLEX
(- mcf) - (H₁(n) . qi) is V31() right_complementable Element of the carrier of F_Complex
(- mcf) + (- (H₁(n) . qi)) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex . ((- mcf),(- (H₁(n) . qi))) is V31() right_complementable Element of the carrier of F_Complex
[(- mcf),(- (H₁(n) . qi))] is non empty set
{(- mcf),(- (H₁(n) . qi))} is non empty finite V67() set
{(- mcf)} is non empty trivial finite 1 -element V67() set
{{(- mcf),(- (H₁(n) . qi))},{(- mcf)}} is non empty finite V44() set
the addF of F_Complex . [(- mcf),(- (H₁(n) . qi))] is set
(- mcf) + (- (H₁(n) . qi)) is V31() Element of COMPLEX
H₁(n) . ps is set
- cb is V31() real ext-real integer rational Element of INT
(n,F_Complex) . 0 is V31() right_complementable Element of the carrier of F_Complex
ps is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Sum ps is V31() right_complementable Element of the carrier of F_Complex
dom ps is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
ps . 1 is set
qi is V31() right_complementable Element of the carrier of F_Complex
<*qi*> is Relation-like NAT -defined the carrier of F_Complex -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of F_Complex *
H₁(n) . 0 is V31() right_complementable Element of the carrier of F_Complex
(fs . 0) * (H₁(n) . 0) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((fs . 0),(H₁(n) . 0)) is V31() right_complementable Element of the carrier of F_Complex
[(fs . 0),(H₁(n) . 0)] is non empty set
{(fs . 0),(H₁(n) . 0)} is non empty finite V67() set
{{(fs . 0),(H₁(n) . 0)},{(fs . 0)}} is non empty finite V44() set
the multF of F_Complex . [(fs . 0),(H₁(n) . 0)] is set
(fs . 0) * (H₁(n) . 0) is V31() Element of COMPLEX
(- (1_ F_Complex)) * (H₁(n) . 0) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((- (1_ F_Complex)),(H₁(n) . 0)) is V31() right_complementable Element of the carrier of F_Complex
[(- (1_ F_Complex)),(H₁(n) . 0)] is non empty set
{(- (1_ F_Complex)),(H₁(n) . 0)} is non empty finite V67() set
{(- (1_ F_Complex))} is non empty trivial finite 1 -element V67() set
{{(- (1_ F_Complex)),(H₁(n) . 0)},{(- (1_ F_Complex))}} is non empty finite V44() set
the multF of F_Complex . [(- (1_ F_Complex)),(H₁(n) . 0)] is set
(- (1_ F_Complex)) * (H₁(n) . 0) is V31() Element of COMPLEX
(1_ F_Complex) * (H₁(n) . 0) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((1_ F_Complex),(H₁(n) . 0)) is V31() right_complementable Element of the carrier of F_Complex
[(1_ F_Complex),(H₁(n) . 0)] is non empty set
{(1_ F_Complex),(H₁(n) . 0)} is non empty finite V67() set
{(1_ F_Complex)} is non empty trivial finite 1 -element V67() set
{{(1_ F_Complex),(H₁(n) . 0)},{(1_ F_Complex)}} is non empty finite V44() set
the multF of F_Complex . [(1_ F_Complex),(H₁(n) . 0)] is set
(1_ F_Complex) * (H₁(n) . 0) is V31() Element of COMPLEX
- ((1_ F_Complex) * (H₁(n) . 0)) is V31() right_complementable Element of the carrier of F_Complex
- (H₁(n) . 0) is V31() right_complementable Element of the carrier of F_Complex
i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) . i is V31() right_complementable Element of the carrier of F_Complex
n is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(n) . 0 is V31() right_complementable Element of the carrier of F_Complex
S is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(S) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(S) . fs is V31() right_complementable Element of the carrier of F_Complex
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
cMGFC is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC),cMGFC) is V31() right_complementable Element of the carrier of F_Complex
len (MGFC) is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
Sum S is V31() right_complementable Element of the carrier of F_Complex
len S is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom S is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S . fs is set
fs -' 1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) . (fs -' 1) is V31() right_complementable Element of the carrier of F_Complex
(power F_Complex) . (cMGFC,(fs -' 1)) is V31() right_complementable Element of the carrier of F_Complex
[cMGFC,(fs -' 1)] is non empty set
{cMGFC,(fs -' 1)} is non empty finite V67() set
{cMGFC} is non empty trivial finite 1 -element V67() set
{{cMGFC,(fs -' 1)},{cMGFC}} is non empty finite V44() set
(power F_Complex) . [cMGFC,(fs -' 1)] is set
((MGFC) . (fs -' 1)) * ((power F_Complex) . (cMGFC,(fs -' 1))) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . (((MGFC) . (fs -' 1)),((power F_Complex) . (cMGFC,(fs -' 1)))) is V31() right_complementable Element of the carrier of F_Complex
[((MGFC) . (fs -' 1)),((power F_Complex) . (cMGFC,(fs -' 1)))] is non empty set
{((MGFC) . (fs -' 1)),((power F_Complex) . (cMGFC,(fs -' 1)))} is non empty finite V67() set
{((MGFC) . (fs -' 1))} is non empty trivial finite 1 -element V67() set
{{((MGFC) . (fs -' 1)),((power F_Complex) . (cMGFC,(fs -' 1)))},{((MGFC) . (fs -' 1))}} is non empty finite V44() set
the multF of F_Complex . [((MGFC) . (fs -' 1)),((power F_Complex) . (cMGFC,(fs -' 1)))] is set
((MGFC) . (fs -' 1)) * ((power F_Complex) . (cMGFC,(fs -' 1))) is V31() Element of COMPLEX
qc is V31() real ext-real integer set
q is V31() real ext-real integer set
qc * q is V31() real ext-real integer rational Element of INT
n is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( ord b₁ divides n & not ord b₁ divides S & not ord b₁ = n ) } is set
Seg n is non empty finite n -element V67() V68() V69() V70() V71() V72() Element of bool NAT
fs is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
Product fs is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) is Relation-like [: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like V25([: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex)) associative Element of bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):]
[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] is Relation-like set
[:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is Relation-like set
bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is set
K228( the carrier of (Polynom-Ring F_Complex),fs, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
q is finite Element of bool the carrier of F_Complex
(q,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
poly_with_roots ((q,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in q & ord b₁ <= a₁ ) } is set
qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in q & ord b₁ <= qc ) } is set
p1 is set
ps is Element of the carrier of (F_Complex)
ord ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in q & ord b₁ <= n ) } is set
len fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 is ordinal natural V31() real ext-real non negative integer finite cardinal set
Seg (len fs) is finite len fs -element V67() V68() V69() V70() V71() V72() Element of bool NAT
Seg p1 is finite p1 -element V67() V68() V69() V70() V71() V72() Element of bool NAT
fs | (Seg p1) is Relation-like NAT -defined Seg p1 -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
[:NAT, the carrier of (Polynom-Ring F_Complex):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring F_Complex):] is non empty non trivial non finite set
ps is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),ps, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
i is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
x is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product x is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),x, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
p1 is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
dom p1 is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in q & ord b₁ <= ps ) } is set
p1 . ps is set
ps + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in q & ord b₁ <= ps + 1 ) } is set
p1 . (ps + 1) is set
Seg ps is finite ps -element V67() V68() V69() V70() V71() V72() Element of bool NAT
fs | (Seg ps) is Relation-like NAT -defined Seg ps -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
qi is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product qi is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),qi, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
Seg (ps + 1) is non empty finite ps + 1 -element K231(ps,1) -element V67() V68() V69() V70() V71() V72() Element of bool NAT
K231(ps,1) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs | (Seg (ps + 1)) is Relation-like NAT -defined Seg (ps + 1) -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
min ((ps + 1),(len fs)) is ordinal natural V31() real ext-real non negative integer finite cardinal Element of REAL
i is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
len i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
x is finite Element of bool the carrier of F_Complex
(x,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((x,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
mc is finite Element of bool the carrier of F_Complex
(mc,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((mc,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
mc is finite Element of bool the carrier of F_Complex
(mc,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((mc,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(fs | (Seg (ps + 1))) . (ps + 1) is set
fs . (ps + 1) is set
i . (ps + 1) is set
jcf is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
Product i is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),i, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
i | (Seg ps) is Relation-like NAT -defined Seg ps -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
<*jcf*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring F_Complex) *
the carrier of (Polynom-Ring F_Complex) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring F_Complex)
qi ^ <*jcf*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
(Product qi) * jcf is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) . ((Product qi),jcf) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
[(Product qi),jcf] is non empty set
{(Product qi),jcf} is non empty finite set
{(Product qi)} is non empty trivial finite 1 -element set
{{(Product qi),jcf},{(Product qi)}} is non empty finite V44() set
the multF of (Polynom-Ring F_Complex) . [(Product qi),jcf] is set
lcf is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(poly_with_roots ((mc,1) -bag)) *' lcf is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
scb is set
cb is Element of the carrier of (F_Complex)
ord cb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
x is Element of the carrier of (F_Complex)
ord x is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cb is Element of the carrier of (F_Complex)
ord cb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ = ps + 1 } is set
((ps + 1)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
scb is non empty finite Element of bool the carrier of F_Complex
(scb,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((scb,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
cb is set
x is Element of the carrier of (F_Complex)
ord x is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
mc \/ scb is non empty finite Element of bool the carrier of F_Complex
x is Element of the carrier of (F_Complex)
ord x is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
x is Element of the carrier of (F_Complex)
ord x is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
mc /\ scb is finite Element of bool the carrier of F_Complex
x is set
fl1pj1i is Element of the carrier of (F_Complex)
ord fl1pj1i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
pi1 is Element of the carrier of (F_Complex)
ord pi1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(poly_with_roots ((mc,1) -bag)) *' (poly_with_roots ((scb,1) -bag)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
((mc,1) -bag) + ((scb,1) -bag) is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support set
poly_with_roots (((mc,1) -bag) + ((scb,1) -bag)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
ps is set
qi is Element of the carrier of (F_Complex)
ord qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qc is finite Element of bool the carrier of F_Complex
qi is Element of the carrier of (F_Complex)
ord qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in q & ord b₁ <= 1 ) } is set
p1 . 1 is set
ps is finite Element of bool the carrier of F_Complex
(ps,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((ps,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
fs | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
i is set
x is Element of the carrier of (F_Complex)
ord x is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
lc is Element of the carrier of (F_Complex)
ord lc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
mc is ordinal natural V31() real ext-real non negative integer finite cardinal set
0 * mc is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered rational V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of NAT
EmptyBag the carrier of F_Complex is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
fs . 1 is set
qi is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
<*(fs . 1)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
i is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
<*i*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring F_Complex) *
the carrier of (Polynom-Ring F_Complex) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring F_Complex)
Product <*i*> is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),<*i*>, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
fs | (Seg (len fs)) is Relation-like NAT -defined Seg (len fs) -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( b₁ in q & ord b₁ <= len fs ) } is set
p1 . (len fs) is set
ps is finite Element of bool the carrier of F_Complex
(ps,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((ps,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
n is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
S is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg n is non empty finite n -element V67() V68() V69() V70() V71() V72() Element of bool NAT
(n,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (n,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
(S,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (S,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
(n) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(S,F_Complex) *' (n) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ( ord b₁ divides n & not ord b₁ divides S & not ord b₁ = n ) } is set
(n) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of n, 1_ F_Complex } is set
q is set
qc is Element of the carrier of (F_Complex)
ord qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q is finite Element of bool the carrier of F_Complex
(q,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
p1 is ordinal natural V31() real ext-real non negative integer finite cardinal set
ps is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(ps) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
p1 is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
dom p1 is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
ps is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 . ps is set
(ps) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
qi is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal set
(qi) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
Product p1 is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) is Relation-like [: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like V25([: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex)) associative Element of bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):]
[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] is Relation-like set
[:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is Relation-like set
bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is set
K228( the carrier of (Polynom-Ring F_Complex),p1, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
poly_with_roots ((q,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
ps is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
((S,F_Complex) *' (n)) *' ps is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
qi is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 . qi is set
(qi) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
((n),1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
(S) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of S, 1_ F_Complex } is set
((S),1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ = n } is set
x is non empty finite Element of bool the carrier of F_Complex
(x,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots ((x,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(S) /\ x is finite Element of bool the carrier of F_Complex
mc is set
jcf is Element of the carrier of (F_Complex)
ord jcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
lcf is Element of the carrier of (F_Complex)
ord lcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(((S),1) -bag) + ((x,1) -bag) is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support set
(S) \/ x is non empty finite Element of bool the carrier of F_Complex
(((S) \/ x),1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
lcf is set
mcf is Element of the carrier of (F_Complex)
ord mcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
jcf is finite Element of bool the carrier of F_Complex
jcf \/ q is finite Element of bool the carrier of F_Complex
jcf is finite Element of bool the carrier of F_Complex
jcf \/ q is finite Element of bool the carrier of F_Complex
jcf is finite Element of bool the carrier of F_Complex
jcf \/ q is finite Element of bool the carrier of F_Complex
jcf is finite Element of bool the carrier of F_Complex
jcf \/ q is finite Element of bool the carrier of F_Complex
jcf is finite Element of bool the carrier of F_Complex
jcf \/ q is finite Element of bool the carrier of F_Complex
mcf is Element of the carrier of (F_Complex)
ord mcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
mcf is Element of the carrier of (F_Complex)
ord mcf is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(jcf,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
poly_with_roots (((S),1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
jcf /\ q is finite Element of bool the carrier of F_Complex
mcf is set
scb is Element of the carrier of (F_Complex)
ord scb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cb is Element of the carrier of (F_Complex)
ord cb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
scb is Element of the carrier of (F_Complex)
ord scb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cb is Element of the carrier of (F_Complex)
ord cb is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
((jcf,1) -bag) + ((q,1) -bag) is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support set
poly_with_roots (((n),1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
poly_with_roots ((jcf,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(poly_with_roots ((jcf,1) -bag)) *' (poly_with_roots ((q,1) -bag)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
((S,F_Complex) *' (n)) *' (poly_with_roots ((q,1) -bag)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
1_. F_Complex is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(0_. F_Complex) +* (0,(1. F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
(1_ F_Complex) * (1_. F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
MGFC is V31() real ext-real integer set
cMGFC is V31() right_complementable Element of the carrier of F_Complex
n is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product n is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) is Relation-like [: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like V25([: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex)) associative Element of bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):]
[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] is Relation-like set
[:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is Relation-like set
bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is set
K228( the carrier of (Polynom-Ring F_Complex),n, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
dom n is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
S is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (S,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
eval ((1_. F_Complex),cMGFC) is V31() right_complementable Element of the carrier of F_Complex
len n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
<*> the carrier of (Polynom-Ring F_Complex) is Relation-like non-empty empty-yielding NAT -defined the carrier of (Polynom-Ring F_Complex) -valued ordinal natural Function-like one-to-one constant functional empty trivial V31() real ext-real non positive non negative integer V38() finite finite-yielding V44() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V57() V58() V59() V60() V61() V62() V63() V64() V67() V68() V69() V70() V71() V72() V73() FinSequence-yielding finite-support Element of the carrier of (Polynom-Ring F_Complex) *
the carrier of (Polynom-Ring F_Complex) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring F_Complex)
1_ (Polynom-Ring F_Complex) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
1. (Polynom-Ring F_Complex) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the OneF of (Polynom-Ring F_Complex) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
len n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg (len n) is finite len n -element V67() V68() V69() V70() V71() V72() Element of bool NAT
n | (Seg (len n)) is Relation-like NAT -defined Seg (len n) -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
[:NAT, the carrier of (Polynom-Ring F_Complex):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring F_Complex):] is non empty non trivial non finite set
fs is ordinal natural V31() real ext-real non negative integer finite cardinal set
Seg fs is finite fs -element V67() V68() V69() V70() V71() V72() Element of bool NAT
n | (Seg fs) is Relation-like NAT -defined Seg fs -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
q is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product q is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),q, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
p1 is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
ps is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),ps, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
fs is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg (q + 1) is non empty finite q + 1 -element K231(q,1) -element V67() V68() V69() V70() V71() V72() Element of bool NAT
K231(q,1) is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n | (Seg (q + 1)) is Relation-like NAT -defined Seg (q + 1) -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
min ((q + 1),(len n)) is ordinal natural V31() real ext-real non negative integer finite cardinal Element of REAL
qc is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
len qc is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Seg q is finite q -element V67() V68() V69() V70() V71() V72() Element of bool NAT
n | (Seg q) is Relation-like NAT -defined Seg q -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
fs . q is set
(n | (Seg (q + 1))) . (q + 1) is set
n . (q + 1) is set
qc . (q + 1) is set
Product qc is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),qc, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
p1 is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product p1 is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),p1, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
qc | (Seg q) is Relation-like NAT -defined Seg q -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
<*ps*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring F_Complex) *
the carrier of (Polynom-Ring F_Complex) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring F_Complex)
p1 ^ <*ps*> is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
(Product p1) * ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) . ((Product p1),ps) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
[(Product p1),ps] is non empty set
{(Product p1),ps} is non empty finite set
{(Product p1)} is non empty trivial finite 1 -element set
{{(Product p1),ps},{(Product p1)}} is non empty finite V44() set
the multF of (Polynom-Ring F_Complex) . [(Product p1),ps] is set
fs . (q + 1) is set
i is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (i,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
x is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (x,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
jcf is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n . jcf is set
jcf is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n . jcf is set
(jcf) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
jcf is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n . jcf is set
(jcf) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
mc is V31() Element of COMPLEX
lc is V31() Element of COMPLEX
mcf is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (mcf,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
qi is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (qi,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
i *' x is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(eval (i,cMGFC)) * (eval (x,cMGFC)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((eval (i,cMGFC)),(eval (x,cMGFC))) is V31() right_complementable Element of the carrier of F_Complex
[(eval (i,cMGFC)),(eval (x,cMGFC))] is non empty set
{(eval (i,cMGFC)),(eval (x,cMGFC))} is non empty finite V67() set
{(eval (i,cMGFC))} is non empty trivial finite 1 -element V67() set
{{(eval (i,cMGFC)),(eval (x,cMGFC))},{(eval (i,cMGFC))}} is non empty finite V44() set
the multF of F_Complex . [(eval (i,cMGFC)),(eval (x,cMGFC))] is set
(eval (i,cMGFC)) * (eval (x,cMGFC)) is V31() Element of COMPLEX
lcf is V31() real ext-real integer set
jcf is V31() real ext-real integer set
lcf * jcf is V31() real ext-real integer rational Element of INT
0 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs . 1 is set
n | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring F_Complex):]
n . 1 is set
qc is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
p1 is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (p1,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
q is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
<*(n . 1)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
Product q is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
K228( the carrier of (Polynom-Ring F_Complex),q, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
fs . (len n) is set
q is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (q,cMGFC) is V31() right_complementable Element of the carrier of F_Complex
len n is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(MGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (MGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
S is V31() real ext-real integer set
cMGFC is V31() real ext-real integer set
n is V31() real ext-real integer set
fs is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC),fs) is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC,F_Complex),fs) is V31() right_complementable Element of the carrier of F_Complex
eval ((1,F_Complex),fs) is V31() right_complementable Element of the carrier of F_Complex
Seg MGFC is non empty finite MGFC -element V67() V68() V69() V70() V71() V72() Element of bool NAT
(1,F_Complex) *' (MGFC) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
qi is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
ps is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product ps is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) is Relation-like [: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like V25([: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex)) associative Element of bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):]
[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] is Relation-like set
[:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is Relation-like set
bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is set
K228( the carrier of (Polynom-Ring F_Complex),ps, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
dom ps is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
((1,F_Complex) *' (MGFC)) *' qi is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (qi,fs) is V31() right_complementable Element of the carrier of F_Complex
x is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
ps . x is set
(x) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
ps . x is set
(x) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
eval (((1,F_Complex) *' (MGFC)),fs) is V31() right_complementable Element of the carrier of F_Complex
(eval (((1,F_Complex) *' (MGFC)),fs)) * (eval (qi,fs)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((eval (((1,F_Complex) *' (MGFC)),fs)),(eval (qi,fs))) is V31() right_complementable Element of the carrier of F_Complex
[(eval (((1,F_Complex) *' (MGFC)),fs)),(eval (qi,fs))] is non empty set
{(eval (((1,F_Complex) *' (MGFC)),fs)),(eval (qi,fs))} is non empty finite V67() set
{(eval (((1,F_Complex) *' (MGFC)),fs))} is non empty trivial finite 1 -element V67() set
{{(eval (((1,F_Complex) *' (MGFC)),fs)),(eval (qi,fs))},{(eval (((1,F_Complex) *' (MGFC)),fs))}} is non empty finite V44() set
the multF of F_Complex . [(eval (((1,F_Complex) *' (MGFC)),fs)),(eval (qi,fs))] is set
(eval (((1,F_Complex) *' (MGFC)),fs)) * (eval (qi,fs)) is V31() Element of COMPLEX
(eval ((1,F_Complex),fs)) * (eval ((MGFC),fs)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((eval ((1,F_Complex),fs)),(eval ((MGFC),fs))) is V31() right_complementable Element of the carrier of F_Complex
[(eval ((1,F_Complex),fs)),(eval ((MGFC),fs))] is non empty set
{(eval ((1,F_Complex),fs)),(eval ((MGFC),fs))} is non empty finite V67() set
{(eval ((1,F_Complex),fs))} is non empty trivial finite 1 -element V67() set
{{(eval ((1,F_Complex),fs)),(eval ((MGFC),fs))},{(eval ((1,F_Complex),fs))}} is non empty finite V44() set
the multF of F_Complex . [(eval ((1,F_Complex),fs)),(eval ((MGFC),fs))] is set
(eval ((1,F_Complex),fs)) * (eval ((MGFC),fs)) is V31() Element of COMPLEX
((eval ((1,F_Complex),fs)) * (eval ((MGFC),fs))) * (eval (qi,fs)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . (((eval ((1,F_Complex),fs)) * (eval ((MGFC),fs))),(eval (qi,fs))) is V31() right_complementable Element of the carrier of F_Complex
[((eval ((1,F_Complex),fs)) * (eval ((MGFC),fs))),(eval (qi,fs))] is non empty set
{((eval ((1,F_Complex),fs)) * (eval ((MGFC),fs))),(eval (qi,fs))} is non empty finite V67() set
{((eval ((1,F_Complex),fs)) * (eval ((MGFC),fs)))} is non empty trivial finite 1 -element V67() set
{{((eval ((1,F_Complex),fs)) * (eval ((MGFC),fs))),(eval (qi,fs))},{((eval ((1,F_Complex),fs)) * (eval ((MGFC),fs)))}} is non empty finite V44() set
the multF of F_Complex . [((eval ((1,F_Complex),fs)) * (eval ((MGFC),fs))),(eval (qi,fs))] is set
((eval ((1,F_Complex),fs)) * (eval ((MGFC),fs))) * (eval (qi,fs)) is V31() Element of COMPLEX
p1 is V31() real ext-real integer set
x is V31() real ext-real integer set
p1 * x is V31() real ext-real integer rational Element of INT
(p1 * x) * cMGFC is V31() real ext-real integer rational Element of INT
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
(MGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (MGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
(cMGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (cMGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
n is V31() real ext-real integer set
(cMGFC,F_Complex) *' (MGFC) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
Seg MGFC is non empty finite MGFC -element V67() V68() V69() V70() V71() V72() Element of bool NAT
q is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
fs is Relation-like NAT -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring F_Complex)
Product fs is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
the multF of (Polynom-Ring F_Complex) is Relation-like [: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] -defined the carrier of (Polynom-Ring F_Complex) -valued Function-like V25([: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex)) associative Element of bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):]
[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):] is Relation-like set
[:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is Relation-like set
bool [:[: the carrier of (Polynom-Ring F_Complex), the carrier of (Polynom-Ring F_Complex):], the carrier of (Polynom-Ring F_Complex):] is set
K228( the carrier of (Polynom-Ring F_Complex),fs, the multF of (Polynom-Ring F_Complex)) is right_complementable Element of the carrier of (Polynom-Ring F_Complex)
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
((cMGFC,F_Complex) *' (MGFC)) *' q is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
qc is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
fs . qc is set
(qc) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
fs . qc is set
(qc) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
qc is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC),qc) is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC,F_Complex),qc) is V31() right_complementable Element of the carrier of F_Complex
eval ((cMGFC,F_Complex),qc) is V31() right_complementable Element of the carrier of F_Complex
eval (q,qc) is V31() right_complementable Element of the carrier of F_Complex
p1 is V31() real ext-real integer set
ps is V31() real ext-real integer set
qi is V31() real ext-real integer set
i is V31() real ext-real integer set
qi div i is V31() real ext-real integer set
x is V31() Element of COMPLEX
lc is V31() Element of COMPLEX
mc is V31() Element of COMPLEX
eval (((cMGFC,F_Complex) *' (MGFC)),qc) is V31() right_complementable Element of the carrier of F_Complex
(eval (((cMGFC,F_Complex) *' (MGFC)),qc)) * (eval (q,qc)) is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((eval (((cMGFC,F_Complex) *' (MGFC)),qc)),(eval (q,qc))) is V31() right_complementable Element of the carrier of F_Complex
[(eval (((cMGFC,F_Complex) *' (MGFC)),qc)),(eval (q,qc))] is non empty set
{(eval (((cMGFC,F_Complex) *' (MGFC)),qc)),(eval (q,qc))} is non empty finite V67() set
{(eval (((cMGFC,F_Complex) *' (MGFC)),qc))} is non empty trivial finite 1 -element V67() set
{{(eval (((cMGFC,F_Complex) *' (MGFC)),qc)),(eval (q,qc))},{(eval (((cMGFC,F_Complex) *' (MGFC)),qc))}} is non empty finite V44() set
the multF of F_Complex . [(eval (((cMGFC,F_Complex) *' (MGFC)),qc)),(eval (q,qc))] is set
(eval (((cMGFC,F_Complex) *' (MGFC)),qc)) * (eval (q,qc)) is V31() Element of COMPLEX
lcf is V31() right_complementable Element of the carrier of F_Complex
jcf is V31() right_complementable Element of the carrier of F_Complex
lcf * jcf is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . (lcf,jcf) is V31() right_complementable Element of the carrier of F_Complex
[lcf,jcf] is non empty set
{lcf,jcf} is non empty finite V67() set
{lcf} is non empty trivial finite 1 -element V67() set
{{lcf,jcf},{lcf}} is non empty finite V44() set
the multF of F_Complex . [lcf,jcf] is set
lcf * jcf is V31() Element of COMPLEX
mcf is V31() right_complementable Element of the carrier of F_Complex
(lcf * jcf) * mcf is V31() right_complementable Element of the carrier of F_Complex
the multF of F_Complex . ((lcf * jcf),mcf) is V31() right_complementable Element of the carrier of F_Complex
[(lcf * jcf),mcf] is non empty set
{(lcf * jcf),mcf} is non empty finite V67() set
{(lcf * jcf)} is non empty trivial finite 1 -element V67() set
{{(lcf * jcf),mcf},{(lcf * jcf)}} is non empty finite V44() set
the multF of F_Complex . [(lcf * jcf),mcf] is set
(lcf * jcf) * mcf is V31() Element of COMPLEX
i * ps is V31() real ext-real integer rational Element of INT
(i * ps) * p1 is V31() real ext-real integer rational Element of INT
ps * p1 is V31() real ext-real integer rational Element of INT
i * (ps * p1) is V31() real ext-real integer rational Element of INT
qi / i is V31() real ext-real Element of REAL
[\(qi / i)/] is V31() real ext-real integer set
(ps * p1) * i is V31() real ext-real integer rational Element of INT
((ps * p1) * i) / i is V31() real ext-real Element of REAL
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
cMGFC |^ MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(cMGFC |^ MGFC) - 1 is V31() real ext-real integer rational Element of INT
n is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC),n) is V31() right_complementable Element of the carrier of F_Complex
(MGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (MGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
S is V31() real ext-real integer set
fs is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC,F_Complex),fs) is V31() right_complementable Element of the carrier of F_Complex
MGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
cMGFC is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
n is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(MGFC) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
n |^ MGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n |^ MGFC) - 1 is V31() real ext-real integer rational Element of INT
n |^ cMGFC is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n |^ cMGFC) - 1 is V31() real ext-real integer rational Element of INT
((n |^ MGFC) - 1) div ((n |^ cMGFC) - 1) is V31() real ext-real integer set
S is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC),S) is V31() right_complementable Element of the carrier of F_Complex
(MGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (MGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
(cMGFC,F_Complex) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support non-zero Element of bool [:NAT, the carrier of F_Complex:]
((0_. F_Complex) +* (0,(- (1_ F_Complex)))) +* (cMGFC,(1_ F_Complex)) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) Element of bool [:NAT, the carrier of F_Complex:]
fs is V31() real ext-real integer set
q is V31() right_complementable Element of the carrier of F_Complex
eval ((MGFC,F_Complex),q) is V31() right_complementable Element of the carrier of F_Complex
qc is V31() right_complementable Element of the carrier of F_Complex
eval ((cMGFC,F_Complex),qc) is V31() right_complementable Element of the carrier of F_Complex
n is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
{ b₁ where b₁ is Element of the carrier of (F_Complex) : ord b₁ = n } is set
S is non empty finite Element of bool the carrier of F_Complex
(S,1) -bag is Relation-like the carrier of F_Complex -defined RAT -valued Function-like total V57() V58() V59() V60() finite-support Element of Bags the carrier of F_Complex
Bags the carrier of F_Complex is non empty set
Bags the carrier of F_Complex is functional non empty Element of bool (Bags the carrier of F_Complex)
bool (Bags the carrier of F_Complex) is set
poly_with_roots ((S,1) -bag) is Relation-like NAT -defined the carrier of F_Complex -valued Function-like V25( NAT , the carrier of F_Complex) finite-Support Element of bool [:NAT, the carrier of F_Complex:]
canFS S is Relation-like NAT -defined S -valued Function-like one-to-one non empty V26(S) finite FinSequence-like FinSubsequence-like finite-support FinSequence of S
rng (canFS S) is non empty finite set
q is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
q - 1 is V31() real ext-real integer rational Element of INT
qc is V31() right_complementable Element of the carrier of F_Complex
eval ((n),qc) is V31() right_complementable Element of the carrier of F_Complex
fs is Relation-like NAT -defined the carrier of F_Complex -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of F_Complex
len fs is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
p1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len p1 is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
dom p1 is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
ps is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(canFS S) . ps is set
p1 . ps is set
qi is V31() right_complementable Element of the carrier of F_Complex
qc - qi is V31() right_complementable Element of the carrier of F_Complex
- qi is V31() right_complementable Element of the carrier of F_Complex
qc + (- qi) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . (qc,(- qi)) is V31() right_complementable Element of the carrier of F_Complex
[qc,(- qi)] is non empty set
{qc,(- qi)} is non empty finite V67() set
{qc} is non empty trivial finite 1 -element V67() set
{{qc,(- qi)},{qc}} is non empty finite V44() set
the addF of F_Complex . [qc,(- qi)] is set
qc + (- qi) is V31() Element of COMPLEX
|.(qc - qi).| is V31() real ext-real Element of REAL
Re (qc - qi) is V31() real ext-real Element of REAL
(Re (qc - qi)) ^2 is V31() real ext-real Element of REAL
K104((Re (qc - qi)),(Re (qc - qi))) is V31() real ext-real set
Im (qc - qi) is V31() real ext-real Element of REAL
(Im (qc - qi)) ^2 is V31() real ext-real Element of REAL
K104((Im (qc - qi)),(Im (qc - qi))) is V31() real ext-real set
((Re (qc - qi)) ^2) + ((Im (qc - qi)) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (qc - qi)) ^2) + ((Im (qc - qi)) ^2)) is V31() real ext-real Element of REAL
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
fs /. ps is V31() right_complementable Element of the carrier of F_Complex
rng p1 is finite set
ps is set
qi is ordinal natural V31() real ext-real non negative integer finite cardinal set
p1 . qi is set
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
fs /. qi is V31() right_complementable Element of the carrier of F_Complex
(canFS S) . qi is set
qc - (fs /. qi) is V31() right_complementable Element of the carrier of F_Complex
- (fs /. qi) is V31() right_complementable Element of the carrier of F_Complex
qc + (- (fs /. qi)) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . (qc,(- (fs /. qi))) is V31() right_complementable Element of the carrier of F_Complex
[qc,(- (fs /. qi))] is non empty set
{qc,(- (fs /. qi))} is non empty finite V67() set
{qc} is non empty trivial finite 1 -element V67() set
{{qc,(- (fs /. qi))},{qc}} is non empty finite V44() set
the addF of F_Complex . [qc,(- (fs /. qi))] is set
qc + (- (fs /. qi)) is V31() Element of COMPLEX
|.(qc - (fs /. qi)).| is V31() real ext-real Element of REAL
Re (qc - (fs /. qi)) is V31() real ext-real Element of REAL
(Re (qc - (fs /. qi))) ^2 is V31() real ext-real Element of REAL
K104((Re (qc - (fs /. qi))),(Re (qc - (fs /. qi)))) is V31() real ext-real set
Im (qc - (fs /. qi)) is V31() real ext-real Element of REAL
(Im (qc - (fs /. qi))) ^2 is V31() real ext-real Element of REAL
K104((Im (qc - (fs /. qi))),(Im (qc - (fs /. qi)))) is V31() real ext-real set
((Re (qc - (fs /. qi))) ^2) + ((Im (qc - (fs /. qi))) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (qc - (fs /. qi))) ^2) + ((Im (qc - (fs /. qi))) ^2)) is V31() real ext-real Element of REAL
card S is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
|.(eval ((n),qc)).| is V31() real ext-real Element of REAL
Re (eval ((n),qc)) is V31() real ext-real Element of REAL
(Re (eval ((n),qc))) ^2 is V31() real ext-real Element of REAL
K104((Re (eval ((n),qc))),(Re (eval ((n),qc)))) is V31() real ext-real set
Im (eval ((n),qc)) is V31() real ext-real Element of REAL
(Im (eval ((n),qc))) ^2 is V31() real ext-real Element of REAL
K104((Im (eval ((n),qc))),(Im (eval ((n),qc)))) is V31() real ext-real set
((Re (eval ((n),qc))) ^2) + ((Im (eval ((n),qc))) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (eval ((n),qc))) ^2) + ((Im (eval ((n),qc))) ^2)) is V31() real ext-real Element of REAL
ps is Relation-like NAT -defined REAL -valued Function-like non empty finite FinSequence-like FinSubsequence-like V57() V58() V59() finite-support FinSequence of REAL
Product ps is V31() real ext-real Element of REAL
rng fs is finite set
dom ps is non empty finite V67() V68() V69() V70() V71() V72() Element of bool NAT
qi is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
ps . qi is V31() real ext-real set
dom fs is finite V67() V68() V69() V70() V71() V72() Element of bool NAT
fs /. qi is V31() right_complementable Element of the carrier of F_Complex
(canFS S) . qi is set
[**q,0**] is V31() right_complementable Element of the carrier of F_Complex
K103(q,K104(0,<i>)) is V31() set
[**q,0**] - (fs /. qi) is V31() right_complementable Element of the carrier of F_Complex
- (fs /. qi) is V31() right_complementable Element of the carrier of F_Complex
[**q,0**] + (- (fs /. qi)) is V31() right_complementable Element of the carrier of F_Complex
the addF of F_Complex is Relation-like [: the carrier of F_Complex, the carrier of F_Complex:] -defined the carrier of F_Complex -valued Function-like V25([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . ([**q,0**],(- (fs /. qi))) is V31() right_complementable Element of the carrier of F_Complex
[[**q,0**],(- (fs /. qi))] is non empty set
{[**q,0**],(- (fs /. qi))} is non empty finite V67() set
{[**q,0**]} is non empty trivial finite 1 -element V67() set
{{[**q,0**],(- (fs /. qi))},{[**q,0**]}} is non empty finite V44() set
the addF of F_Complex . [[**q,0**],(- (fs /. qi))] is set
[**q,0**] + (- (fs /. qi)) is V31() Element of COMPLEX
|.([**q,0**] - (fs /. qi)).| is V31() real ext-real Element of REAL
Re ([**q,0**] - (fs /. qi)) is V31() real ext-real Element of REAL
(Re ([**q,0**] - (fs /. qi))) ^2 is V31() real ext-real Element of REAL
K104((Re ([**q,0**] - (fs /. qi))),(Re ([**q,0**] - (fs /. qi)))) is V31() real ext-real set
Im ([**q,0**] - (fs /. qi)) is V31() real ext-real Element of REAL
(Im ([**q,0**] - (fs /. qi))) ^2 is V31() real ext-real Element of REAL
K104((Im ([**q,0**] - (fs /. qi))),(Im ([**q,0**] - (fs /. qi)))) is V31() real ext-real set
((Re ([**q,0**] - (fs /. qi))) ^2) + ((Im ([**q,0**] - (fs /. qi))) ^2) is V31() real ext-real Element of REAL
sqrt (((Re ([**q,0**] - (fs /. qi))) ^2) + ((Im ([**q,0**] - (fs /. qi))) ^2)) is V31() real ext-real Element of REAL
fs . qi is set
1_ (F_Complex) is non being_of_order_0 Element of the carrier of (F_Complex)
i is Element of the carrier of (F_Complex)
ord i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
(n) is non empty finite Element of bool the carrier of F_Complex
{ b₁ where b₁ is V31() right_complementable Element of the carrier of F_Complex : b₁ is V31() CRoot of n, 1_ F_Complex } is set
i is Element of the carrier of (F_Complex)
ord i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
|.(fs /. qi).| is V31() real ext-real Element of REAL
Re (fs /. qi) is V31() real ext-real Element of REAL
(Re (fs /. qi)) ^2 is V31() real ext-real Element of REAL
K104((Re (fs /. qi)),(Re (fs /. qi))) is V31() real ext-real set
Im (fs /. qi) is V31() real ext-real Element of REAL
(Im (fs /. qi)) ^2 is V31() real ext-real Element of REAL
K104((Im (fs /. qi)),(Im (fs /. qi))) is V31() real ext-real set
((Re (fs /. qi)) ^2) + ((Im (fs /. qi)) ^2) is V31() real ext-real Element of REAL
sqrt (((Re (fs /. qi)) ^2) + ((Im (fs /. qi)) ^2)) is V31() real ext-real Element of REAL
1 + 1 is ordinal natural non empty V31() real ext-real positive non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
qi is V31() real ext-real integer set
(1 + 1) + (- 1) is V31() real ext-real integer rational Element of INT
qi + (- 1) is V31() real ext-real integer rational Element of INT
i is V31() real ext-real integer set
abs i is ordinal natural V31() real ext-real non negative integer finite cardinal rational V67() V68() V69() V70() V71() V72() Element of NAT
Re i is V31() real ext-real Element of REAL
(Re i) ^2 is V31() real ext-real Element of REAL
K104((Re i),(Re i)) is V31() real ext-real set
Im i is V31() real ext-real Element of REAL
(Im i) ^2 is V31() real ext-real Element of REAL
K104((Im i),(Im i)) is V31() real ext-real set
((Re i) ^2) + ((Im i) ^2) is V31() real ext-real Element of REAL
sqrt (((Re i) ^2) + ((Im i) ^2)) is V31() real ext-real Element of REAL
x is V31() real ext-real Element of REAL