:: Fundamental Theorem of Algebra
:: by Robert Milewski
::
:: Received August 21, 2000
:: Copyright (c) 2000-2011 Association of Mizar Users


begin

theorem Th1: :: POLYNOM5:1
for n, m being Element of NAT st n <> 0 & m <> 0 holds
(((n * m) - n) - m) + 1 >= 0
proof end;

theorem Th2: :: POLYNOM5:2
for x, y being real number st y > 0 holds
(min (x,y)) / (max (x,y)) <= 1
proof end;

theorem Th3: :: POLYNOM5:3
for x, y being real number st ( for c being real number st c > 0 & c < 1 holds
c * x >= y ) holds
y <= 0
proof end;

Lm1: for x, y being Real st ( for c being Real st c > 0 & c < 1 holds
c * x >= y ) holds
y <= 0
proof end;

theorem Th4: :: POLYNOM5:4
for p being FinSequence of REAL st ( for n being Element of NAT st n in dom p holds
p . n >= 0 ) holds
for i being Element of NAT st i in dom p holds
Sum p >= p . i
proof end;

theorem Th5: :: POLYNOM5:5
for x, y being Real holds - [**x,y**] = [**(- x),(- y)**]
proof end;

theorem Th6: :: POLYNOM5:6
for x1, y1, x2, y2 being Real holds [**x1,y1**] - [**x2,y2**] = [**(x1 - x2),(y1 - y2)**]
proof end;

theorem :: POLYNOM5:7
canceled;

theorem Th8: :: POLYNOM5:8
for z being Element of F_Complex st z <> 0. F_Complex holds
for n being Element of NAT holds |.((power F_Complex) . (z,n)).| = |.z.| to_power n
proof end;

definition
let p be FinSequence of the carrier of F_Complex;
func |.p.| -> FinSequence of REAL means :Def1: :: POLYNOM5:def 1
( len it = len p & ( for n being Element of NAT st n in dom p holds
it /. n = |.(p /. n).| ) );
existence
ex b1 being FinSequence of REAL st
( len b1 = len p & ( for n being Element of NAT st n in dom p holds
b1 /. n = |.(p /. n).| ) )
proof end;
uniqueness
for b1, b2 being FinSequence of REAL st len b1 = len p & ( for n being Element of NAT st n in dom p holds
b1 /. n = |.(p /. n).| ) & len b2 = len p & ( for n being Element of NAT st n in dom p holds
b2 /. n = |.(p /. n).| ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines |. POLYNOM5:def 1 :
for p being FinSequence of the carrier of F_Complex
for b2 being FinSequence of REAL holds
( b2 = |.p.| iff ( len b2 = len p & ( for n being Element of NAT st n in dom p holds
b2 /. n = |.(p /. n).| ) ) );

theorem Th9: :: POLYNOM5:9
|.(<*> the carrier of F_Complex).| = <*> REAL
proof end;

theorem Th10: :: POLYNOM5:10
for x being Element of F_Complex holds |.<*x*>.| = <*|.x.|*>
proof end;

theorem :: POLYNOM5:11
for x, y being Element of F_Complex holds |.<*x,y*>.| = <*|.x.|,|.y.|*>
proof end;

theorem :: POLYNOM5:12
for x, y, z being Element of F_Complex holds |.<*x,y,z*>.| = <*|.x.|,|.y.|,|.z.|*>
proof end;

theorem Th13: :: POLYNOM5:13
for p, q being FinSequence of the carrier of F_Complex holds |.(p ^ q).| = |.p.| ^ |.q.|
proof end;

theorem :: POLYNOM5:14
for p being FinSequence of the carrier of F_Complex
for x being Element of F_Complex holds
( |.(p ^ <*x*>).| = |.p.| ^ <*|.x.|*> & |.(<*x*> ^ p).| = <*|.x.|*> ^ |.p.| )
proof end;

theorem Th15: :: POLYNOM5:15
for p being FinSequence of the carrier of F_Complex holds |.(Sum p).| <= Sum |.p.|
proof end;

begin

definition
let L be non empty right_complementable Abelian add-associative right_zeroed commutative right_unital distributive doubleLoopStr ;
let p be Polynomial of L;
let n be Element of NAT ;
func p `^ n -> sequence of L equals :: POLYNOM5:def 2
(power (Polynom-Ring L)) . (p,n);
coherence
(power (Polynom-Ring L)) . (p,n) is sequence of L
proof end;
end;

:: deftheorem defines `^ POLYNOM5:def 2 :
for L being non empty right_complementable Abelian add-associative right_zeroed commutative right_unital distributive doubleLoopStr
for p being Polynomial of L
for n being Element of NAT holds p `^ n = (power (Polynom-Ring L)) . (p,n);

registration
let L be non empty right_complementable Abelian add-associative right_zeroed commutative right_unital distributive doubleLoopStr ;
let p be Polynomial of L;
let n be Element of NAT ;
cluster p `^ n -> finite-Support ;
coherence
p `^ n is finite-Support
proof end;
end;

theorem Th16: :: POLYNOM5:16
for L being non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr
for p being Polynomial of L holds p `^ 0 = 1_. L
proof end;

theorem :: POLYNOM5:17
for L being non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr
for p being Polynomial of L holds p `^ 1 = p
proof end;

theorem :: POLYNOM5:18
for L being non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr
for p being Polynomial of L holds p `^ 2 = p *' p
proof end;

theorem :: POLYNOM5:19
for L being non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr
for p being Polynomial of L holds p `^ 3 = (p *' p) *' p
proof end;

theorem Th20: :: POLYNOM5:20
for L being non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr
for p being Polynomial of L
for n being Element of NAT holds p `^ (n + 1) = (p `^ n) *' p
proof end;

theorem Th21: :: POLYNOM5:21
for L being non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr
for n being Element of NAT holds (0_. L) `^ (n + 1) = 0_. L
proof end;

theorem :: POLYNOM5:22
for L being non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr
for n being Element of NAT holds (1_. L) `^ n = 1_. L
proof end;

theorem Th23: :: POLYNOM5:23
for L being Field
for p being Polynomial of L
for x being Element of L
for n being Element of NAT holds eval ((p `^ n),x) = (power L) . ((eval (p,x)),n)
proof end;

Lm2: for L being non empty ZeroStr
for p being AlgSequence of L st len p > 0 holds
p . ((len p) -' 1) <> 0. L
proof end;

theorem Th24: :: POLYNOM5:24
for L being domRing
for p being Polynomial of L st len p <> 0 holds
for n being Element of NAT holds len (p `^ n) = ((n * (len p)) - n) + 1
proof end;

definition
let L be non empty multMagma ;
let p be sequence of L;
let v be Element of L;
func v * p -> sequence of L means :Def3: :: POLYNOM5:def 3
 for n being Element of NAT holds it . n = v * (p . n);
existence
ex b1 being sequence of L st
for n being Element of NAT holds b1 . n = v * (p . n)
proof end;
uniqueness
for b1, b2 being sequence of L st ( for n being Element of NAT holds b1 . n = v * (p . n) ) & ( for n being Element of NAT holds b2 . n = v * (p . n) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines * POLYNOM5:def 3 :
for L being non empty multMagma
for p being sequence of L
for v being Element of L
for b4 being sequence of L holds
( b4 = v * p iff for n being Element of NAT holds b4 . n = v * (p . n) );

registration
let L be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr ;
let p be Polynomial of L;
let v be Element of L;
cluster v * p -> finite-Support ;
coherence
v * p is finite-Support
proof end;
end;

theorem Th25: :: POLYNOM5:25
for L being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr
for p being Polynomial of L holds len ((0. L) * p) = 0
proof end;

theorem Th26: :: POLYNOM5:26
for L being non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for p being Polynomial of L
for v being Element of L st v <> 0. L holds
len (v * p) = len p
proof end;

theorem Th27: :: POLYNOM5:27
for L being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr
for p being sequence of L holds (0. L) * p = 0_. L
proof end;

theorem Th28: :: POLYNOM5:28
for L being non empty well-unital multLoopStr
for p being sequence of L holds (1. L) * p = p
proof end;

theorem Th29: :: POLYNOM5:29
for L being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for v being Element of L holds v * (0_. L) = 0_. L
proof end;

theorem Th30: :: POLYNOM5:30
for L being non empty right_complementable add-associative right_zeroed right-distributive well-unital doubleLoopStr
for v being Element of L holds v * (1_. L) = <%v%>
proof end;

theorem Th31: :: POLYNOM5:31
for L being non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for p being Polynomial of L
for v, x being Element of L holds eval ((v * p),x) = v * (eval (p,x))
proof end;

theorem Th32: :: POLYNOM5:32
for L being non empty right_complementable add-associative right_zeroed unital right-distributive doubleLoopStr
for p being Polynomial of L holds eval (p,(0. L)) = p . 0
proof end;

definition
let L be non empty ZeroStr ;
let z0, z1 be Element of L;
func <%z0,z1%> -> sequence of L equals :: POLYNOM5:def 4
((0_. L) +* (0,z0)) +* (1,z1);
coherence
((0_. L) +* (0,z0)) +* (1,z1) is sequence of L ;
end;

:: deftheorem defines <% POLYNOM5:def 4 :
for L being non empty ZeroStr
for z0, z1 being Element of L holds <%z0,z1%> = ((0_. L) +* (0,z0)) +* (1,z1);

theorem Th33: :: POLYNOM5:33
for L being non empty ZeroStr
for z0 being Element of L holds
( <%z0%> . 0 = z0 & ( for n being Element of NAT st n >= 1 holds
<%z0%> . n = 0. L ) )
proof end;

theorem Th34: :: POLYNOM5:34
for L being non empty ZeroStr
for z0 being Element of L st z0 <> 0. L holds
len <%z0%> = 1
proof end;

theorem Th35: :: POLYNOM5:35
for L being non empty ZeroStr holds <%(0. L)%> = 0_. L
proof end;

theorem Th36: :: POLYNOM5:36
for L being non empty right_complementable add-associative right_zeroed associative commutative well-unital distributive domRing-like doubleLoopStr
for x, y being Element of L holds <%x%> *' <%y%> = <%(x * y)%>
proof end;

theorem Th37: :: POLYNOM5:37
for L being non empty right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for x being Element of L
for n being Element of NAT holds <%x%> `^ n = <%((power L) . (x,n))%>
proof end;

theorem :: POLYNOM5:38
for L being non empty right_complementable add-associative right_zeroed unital doubleLoopStr
for z0, x being Element of L holds eval (<%z0%>,x) = z0
proof end;

theorem Th39: :: POLYNOM5:39
for L being non empty ZeroStr
for z0, z1 being Element of L holds
( <%z0,z1%> . 0 = z0 & <%z0,z1%> . 1 = z1 & ( for n being Nat st n >= 2 holds
<%z0,z1%> . n = 0. L ) )
proof end;

registration
let L be non empty ZeroStr ;
let z0, z1 be Element of L;
cluster <%z0,z1%> -> finite-Support ;
coherence
<%z0,z1%> is finite-Support
proof end;
end;

theorem Th40: :: POLYNOM5:40
for L being non empty ZeroStr
for z0, z1 being Element of L holds len <%z0,z1%> <= 2
proof end;

theorem Th41: :: POLYNOM5:41
for L being non empty ZeroStr
for z0, z1 being Element of L st z1 <> 0. L holds
len <%z0,z1%> = 2
proof end;

theorem Th42: :: POLYNOM5:42
for L being non empty ZeroStr
for z0 being Element of L st z0 <> 0. L holds
len <%z0,(0. L)%> = 1
proof end;

theorem Th43: :: POLYNOM5:43
for L being non empty ZeroStr holds <%(0. L),(0. L)%> = 0_. L
proof end;

theorem :: POLYNOM5:44
for L being non empty ZeroStr
for z0 being Element of L holds <%z0,(0. L)%> = <%z0%>
proof end;

theorem Th45: :: POLYNOM5:45
for L being non empty right_complementable add-associative right_zeroed unital left-distributive doubleLoopStr
for z0, z1, x being Element of L holds eval (<%z0,z1%>,x) = z0 + (z1 * x)
proof end;

theorem :: POLYNOM5:46
for L being non empty right_complementable add-associative right_zeroed left-distributive well-unital doubleLoopStr
for z0, z1, x being Element of L holds eval (<%z0,(0. L)%>,x) = z0
proof end;

theorem :: POLYNOM5:47
for L being non empty right_complementable add-associative right_zeroed unital left-distributive doubleLoopStr
for z0, z1, x being Element of L holds eval (<%(0. L),z1%>,x) = z1 * x
proof end;

theorem Th48: :: POLYNOM5:48
for L being non empty right_complementable add-associative right_zeroed left-distributive well-unital doubleLoopStr
for z0, z1, x being Element of L holds eval (<%z0,(1. L)%>,x) = z0 + x
proof end;

theorem :: POLYNOM5:49
for L being non empty right_complementable add-associative right_zeroed left-distributive well-unital doubleLoopStr
for z0, z1, x being Element of L holds eval (<%(0. L),(1. L)%>,x) = x
proof end;

begin

definition
let L be non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr ;
let p, q be Polynomial of L;
func Subst (p,q) -> Polynomial of L means :Def5: :: POLYNOM5:def 5
 ex F being FinSequence of the carrier of (Polynom-Ring L) st
( it = Sum F & len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = (p . (n -' 1)) * (q `^ (n -' 1)) ) );
existence
ex b1 being Polynomial of L ex F being FinSequence of the carrier of (Polynom-Ring L) st
( b1 = Sum F & len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = (p . (n -' 1)) * (q `^ (n -' 1)) ) )
proof end;
uniqueness
for b1, b2 being Polynomial of L st ex F being FinSequence of the carrier of (Polynom-Ring L) st
( b1 = Sum F & len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = (p . (n -' 1)) * (q `^ (n -' 1)) ) ) & ex F being FinSequence of the carrier of (Polynom-Ring L) st
( b2 = Sum F & len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = (p . (n -' 1)) * (q `^ (n -' 1)) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines Subst POLYNOM5:def 5 :
for L being non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr
for p, q, b4 being Polynomial of L holds
( b4 = Subst (p,q) iff ex F being FinSequence of the carrier of (Polynom-Ring L) st
( b4 = Sum F & len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = (p . (n -' 1)) * (q `^ (n -' 1)) ) ) );

theorem Th50: :: POLYNOM5:50
for L being non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr
for p being Polynomial of L holds Subst ((0_. L),p) = 0_. L
proof end;

theorem :: POLYNOM5:51
for L being non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive doubleLoopStr
for p being Polynomial of L holds Subst (p,(0_. L)) = <%(p . 0)%>
proof end;

theorem :: POLYNOM5:52
for L being non empty right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for p being Polynomial of L
for x being Element of L holds len (Subst (p,<%x%>)) <= 1
proof end;

theorem Th53: :: POLYNOM5:53
for L being Field
for p, q being Polynomial of L st len p <> 0 & len q > 1 holds
len (Subst (p,q)) = ((((len p) * (len q)) - (len p)) - (len q)) + 2
proof end;

theorem Th54: :: POLYNOM5:54
for L being Field
for p, q being Polynomial of L
for x being Element of L holds eval ((Subst (p,q)),x) = eval (p,(eval (q,x)))
proof end;

begin

definition
let L be non empty unital doubleLoopStr ;
let p be Polynomial of L;
let x be Element of L;
pred x is_a_root_of p means :Def6: :: POLYNOM5:def 6
 eval (p,x) = 0. L;
end;

:: deftheorem Def6 defines is_a_root_of POLYNOM5:def 6 :
for L being non empty unital doubleLoopStr
for p being Polynomial of L
for x being Element of L holds
( x is_a_root_of p iff eval (p,x) = 0. L );

definition
let L be non empty unital doubleLoopStr ;
let p be Polynomial of L;
attr p is with_roots means :Def7: :: POLYNOM5:def 7
 ex x being Element of L st x is_a_root_of p;
end;

:: deftheorem Def7 defines with_roots POLYNOM5:def 7 :
for L being non empty unital doubleLoopStr
for p being Polynomial of L holds
( p is with_roots iff ex x being Element of L st x is_a_root_of p );

theorem Th55: :: POLYNOM5:55
for L being non empty unital doubleLoopStr holds 0_. L is with_roots
proof end;

registration
let L be non empty unital doubleLoopStr ;
cluster 0_. L -> with_roots ;
coherence
0_. L is with_roots by Th55;
end;

theorem :: POLYNOM5:56
for L being non empty unital doubleLoopStr
for x being Element of L holds x is_a_root_of 0_. L
proof end;

registration
let L be non empty unital doubleLoopStr ;
cluster Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of K29(K30(NAT, the carrier of L));
existence
ex b1 being Polynomial of L st b1 is with_roots
proof end;
end;

definition
let L be non empty unital doubleLoopStr ;
attr L is algebraic-closed means :Def8: :: POLYNOM5:def 8
 for p being Polynomial of L st len p > 1 holds
p is with_roots ;
end;

:: deftheorem Def8 defines algebraic-closed POLYNOM5:def 8 :
for L being non empty unital doubleLoopStr holds
( L is algebraic-closed iff for p being Polynomial of L st len p > 1 holds
p is with_roots );

definition
let L be non empty unital doubleLoopStr ;
let p be Polynomial of L;
func Roots p -> Subset of L means :Def9: :: POLYNOM5:def 9
 for x being Element of L holds
( x in it iff x is_a_root_of p );
existence
ex b1 being Subset of L st
for x being Element of L holds
( x in b1 iff x is_a_root_of p )
proof end;
uniqueness
for b1, b2 being Subset of L st ( for x being Element of L holds
( x in b1 iff x is_a_root_of p ) ) & ( for x being Element of L holds
( x in b2 iff x is_a_root_of p ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def9 defines Roots POLYNOM5:def 9 :
for L being non empty unital doubleLoopStr
for p being Polynomial of L
for b3 being Subset of L holds
( b3 = Roots p iff for x being Element of L holds
( x in b3 iff x is_a_root_of p ) );

definition
let L be non empty almost_left_invertible associative commutative well-unital distributive doubleLoopStr ;
let p be Polynomial of L;
func NormPolynomial p -> sequence of L means :Def10: :: POLYNOM5:def 10
 for n being Element of NAT holds it . n = (p . n) / (p . ((len p) -' 1));
existence
ex b1 being sequence of L st
for n being Element of NAT holds b1 . n = (p . n) / (p . ((len p) -' 1))
proof end;
uniqueness
for b1, b2 being sequence of L st ( for n being Element of NAT holds b1 . n = (p . n) / (p . ((len p) -' 1)) ) & ( for n being Element of NAT holds b2 . n = (p . n) / (p . ((len p) -' 1)) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def10 defines NormPolynomial POLYNOM5:def 10 :
for L being non empty almost_left_invertible associative commutative well-unital distributive doubleLoopStr
for p being Polynomial of L
for b3 being sequence of L holds
( b3 = NormPolynomial p iff for n being Element of NAT holds b3 . n = (p . n) / (p . ((len p) -' 1)) );

registration
let L be non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr ;
let p be Polynomial of L;
cluster NormPolynomial p -> finite-Support ;
coherence
NormPolynomial p is finite-Support
proof end;
end;

theorem Th57: :: POLYNOM5:57
for L being non empty almost_left_invertible associative commutative well-unital distributive doubleLoopStr
for p being Polynomial of L st len p <> 0 holds
(NormPolynomial p) . ((len p) -' 1) = 1. L
proof end;

theorem Th58: :: POLYNOM5:58
for L being Field
for p being Polynomial of L st len p <> 0 holds
len (NormPolynomial p) = len p
proof end;

theorem Th59: :: POLYNOM5:59
for L being Field
for p being Polynomial of L st len p <> 0 holds
for x being Element of L holds eval ((NormPolynomial p),x) = (eval (p,x)) / (p . ((len p) -' 1))
proof end;

theorem Th60: :: POLYNOM5:60
for L being Field
for p being Polynomial of L st len p <> 0 holds
for x being Element of L holds
( x is_a_root_of p iff x is_a_root_of NormPolynomial p )
proof end;

theorem Th61: :: POLYNOM5:61
for L being Field
for p being Polynomial of L st len p <> 0 holds
( p is with_roots iff NormPolynomial p is with_roots )
proof end;

theorem :: POLYNOM5:62
for L being Field
for p being Polynomial of L st len p <> 0 holds
Roots p = Roots (NormPolynomial p)
proof end;

theorem Th63: :: POLYNOM5:63
id COMPLEX is_continuous_on COMPLEX
proof end;

theorem Th64: :: POLYNOM5:64
for x being Element of COMPLEX holds COMPLEX --> x is_continuous_on COMPLEX
proof end;

definition
let L be non empty unital multMagma ;
let x be Element of L;
let n be Element of NAT ;
func FPower (x,n) -> Function of L,L means :Def11: :: POLYNOM5:def 11
 for y being Element of L holds it . y = x * ((power L) . (y,n));
existence
ex b1 being Function of L,L st
for y being Element of L holds b1 . y = x * ((power L) . (y,n))
proof end;
uniqueness
for b1, b2 being Function of L,L st ( for y being Element of L holds b1 . y = x * ((power L) . (y,n)) ) & ( for y being Element of L holds b2 . y = x * ((power L) . (y,n)) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines FPower POLYNOM5:def 11 :
for L being non empty unital multMagma
for x being Element of L
for n being Element of NAT
for b4 being Function of L,L holds
( b4 = FPower (x,n) iff for y being Element of L holds b4 . y = x * ((power L) . (y,n)) );

theorem :: POLYNOM5:65
for L being non empty unital multMagma holds FPower ((1_ L),1) = id the carrier of L
proof end;

theorem :: POLYNOM5:66
FPower ((1_ F_Complex),2) = (id COMPLEX) (#) (id COMPLEX)
proof end;

theorem Th67: :: POLYNOM5:67
for L being non empty unital multMagma
for x being Element of L holds FPower (x,0) = the carrier of L --> x
proof end;

theorem :: POLYNOM5:68
for x being Element of F_Complex ex x1 being Element of COMPLEX st
( x = x1 & FPower (x,1) = x1 (#) (id COMPLEX) )
proof end;

theorem :: POLYNOM5:69
for x being Element of F_Complex ex x1 being Element of COMPLEX st
( x = x1 & FPower (x,2) = x1 (#) ((id COMPLEX) (#) (id COMPLEX)) )
proof end;

theorem Th70: :: POLYNOM5:70
for x being Element of F_Complex
for n being Element of NAT ex f being Function of COMPLEX,COMPLEX st
( f = FPower (x,n) & FPower (x,(n + 1)) = f (#) (id COMPLEX) )
proof end;

theorem Th71: :: POLYNOM5:71
for x being Element of F_Complex
for n being Element of NAT ex f being Function of COMPLEX,COMPLEX st
( f = FPower (x,n) & f is_continuous_on COMPLEX )
proof end;

definition
let L be non empty well-unital doubleLoopStr ;
let p be Polynomial of L;
func Polynomial-Function (L,p) -> Function of L,L means :Def12: :: POLYNOM5:def 12
 for x being Element of L holds it . x = eval (p,x);
existence
ex b1 being Function of L,L st
for x being Element of L holds b1 . x = eval (p,x)
proof end;
uniqueness
for b1, b2 being Function of L,L st ( for x being Element of L holds b1 . x = eval (p,x) ) & ( for x being Element of L holds b2 . x = eval (p,x) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines Polynomial-Function POLYNOM5:def 12 :
for L being non empty well-unital doubleLoopStr
for p being Polynomial of L
for b3 being Function of L,L holds
( b3 = Polynomial-Function (L,p) iff for x being Element of L holds b3 . x = eval (p,x) );

theorem Th72: :: POLYNOM5:72
for p being Polynomial of F_Complex ex f being Function of COMPLEX,COMPLEX st
( f = Polynomial-Function (F_Complex,p) & f is_continuous_on COMPLEX )
proof end;

theorem Th73: :: POLYNOM5:73
for p being Polynomial of F_Complex st len p > 2 & |.(p . ((len p) -' 1)).| = 1 holds
for F being FinSequence of REAL st len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = |.(p . (n -' 1)).| ) holds
for z being Element of F_Complex st |.z.| > Sum F holds
|.(eval (p,z)).| > |.(p . 0).| + 1
proof end;

theorem Th74: :: POLYNOM5:74
for p being Polynomial of F_Complex st len p > 2 holds
ex z0 being Element of F_Complex st
for z being Element of F_Complex holds |.(eval (p,z)).| >= |.(eval (p,z0)).|
proof end;

theorem Th75: :: POLYNOM5:75
for p being Polynomial of F_Complex st len p > 1 holds
p is with_roots
proof end;

registration
cluster F_Complex -> algebraic-closed ;
coherence
F_Complex is algebraic-closed by Def8, Th75;
end;

registration
cluster non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right_unital well-unital distributive left_unital algebraic-closed doubleLoopStr ;
existence
ex b1 being non empty well-unital doubleLoopStr st
( b1 is algebraic-closed & b1 is add-associative & b1 is right_zeroed & b1 is right_complementable & b1 is Abelian & b1 is commutative & b1 is associative & b1 is distributive & b1 is almost_left_invertible & not b1 is degenerated )
proof end;
end;