POLYNOM5

:: POLYNOM5 semantic presentation

begin

for n, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) holds
(((n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) * m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) - n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( ) ( complex real ext-real V54() ) Element of REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) - m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( ) ( complex real ext-real V54() ) Element of REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) + 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( complex real ext-real V54() ) Element of REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) >= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ;

theorem :: POLYNOM5:2

for x, y being ( ( real ) ( complex real ext-real ) number ) st y : ( ( real ) ( complex real ext-real ) number ) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) holds
(min (x : ( ( real ) ( complex real ext-real ) number ) ,y : ( ( real ) ( complex real ext-real ) number ) )) : ( ( ) ( complex real ext-real ) set ) / (max (x : ( ( real ) ( complex real ext-real ) number ) ,y : ( ( real ) ( complex real ext-real ) number ) )) : ( ( ) ( complex real ext-real ) set ) : ( ( ) ( complex real ext-real ) Element of COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) <= 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ;

theorem :: POLYNOM5:3

for x, y being ( ( real ) ( complex real ext-real ) number ) st ( for c being ( ( real ) ( complex real ext-real ) number ) st c : ( ( real ) ( complex real ext-real ) number ) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) & c : ( ( real ) ( complex real ext-real ) number ) < 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) holds
c : ( ( real ) ( complex real ext-real ) number ) * x : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) set ) >= y : ( ( real ) ( complex real ext-real ) number ) ) holds
y : ( ( real ) ( complex real ext-real ) number ) <= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ;

theorem :: POLYNOM5:4

theorem :: POLYNOM5:5

for x, y being ( ( ) ( complex real ext-real ) Real) holds - [**x : ( ( ) ( complex real ext-real ) Real) ,y : ( ( ) ( complex real ext-real ) Real) **] : ( ( ) ( complex ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) ) : ( ( ) ( complex ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) ) = [**(- x : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) ,(- y : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) **] : ( ( ) ( complex ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) ) ;

theorem :: POLYNOM5:6

theorem :: POLYNOM5:7

definition

let p be ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) -valued Function-like V24() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) ) ;

func |.p.| -> ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) -valued Function-like V24() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) means :: POLYNOM5:def 1
( len it : ( ( ) ( ) VectSpStr over p : ( ( ) ( ) 1-sorted ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) = len p : ( ( ) ( ) 1-sorted ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) in dom p : ( ( ) ( ) 1-sorted ) : ( ( ) ( V56() V57() V58() V59() V60() V61() ) Element of K27(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( V24() ) set ) ) holds
it : ( ( ) ( ) VectSpStr over p : ( ( ) ( ) 1-sorted ) ) /. n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) = |.(p : ( ( ) ( ) 1-sorted ) /. n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( ) ( complex ) Element of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) ) );

end;

theorem :: POLYNOM5:8

|.(<*> the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty V24() V29() V31( {} : ( ( ) ( ) set ) ) FinSequence-like FinSubsequence-like FinSequence-membered complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V54() V56() V57() V58() V59() V60() V61() V62() Function-yielding V87() ) FinSequence of the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) ) .| : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) -valued Function-like V24() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) = <*> REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty V24() V29() V31( {} : ( ( ) ( ) set ) ) FinSequence-like FinSubsequence-like FinSequence-membered complex real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V54() V56() V57() V58() V59() V60() V61() V62() Function-yielding V87() ) FinSequence of REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) ;

theorem :: POLYNOM5:9

theorem :: POLYNOM5:10

theorem :: POLYNOM5:11

theorem :: POLYNOM5:12

theorem :: POLYNOM5:13

theorem :: POLYNOM5:14

begin

definition

let L be ( ( non empty right_complementable Abelian add-associative right_zeroed commutative right_unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable Abelian add-associative right_zeroed commutative right-distributive left-distributive right_unital distributive V174() V175() V176() V177() ) doubleLoopStr ) ;
let p be ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty right_complementable Abelian add-associative right_zeroed commutative right_unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable Abelian add-associative right_zeroed commutative right-distributive left-distributive right_unital distributive V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Polynomial of L : ( ( non empty right_complementable Abelian add-associative right_zeroed commutative right_unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable Abelian add-associative right_zeroed commutative right-distributive left-distributive right_unital distributive V174() V175() V176() V177() ) doubleLoopStr ) ) ;
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ;

func p `^ n -> ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( ) set ) ) equals :: POLYNOM5:def 2
(power (Polynom-Ring L : ( ( ) ( ) 1-sorted ) ) : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( Relation-like K28( the carrier of (Polynom-Ring L : ( ( ) ( ) 1-sorted ) ) : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ,NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V24() V56() V57() V58() V59() V62() ) set ) -valued INT : ( ( ) ( non empty V24() V56() V57() V58() V59() V60() V62() ) set ) -valued V24() complex-valued ext-real-valued real-valued natural-valued ) set ) -defined the carrier of (Polynom-Ring L : ( ( ) ( ) 1-sorted ) ) : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K27(K28(K28( the carrier of (Polynom-Ring L : ( ( ) ( ) 1-sorted ) ) : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ,NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V24() V56() V57() V58() V59() V62() ) set ) -valued INT : ( ( ) ( non empty V24() V56() V57() V58() V59() V60() V62() ) set ) -valued V24() complex-valued ext-real-valued real-valued natural-valued ) set ) , the carrier of (Polynom-Ring L : ( ( ) ( ) 1-sorted ) ) : ( ( non empty strict ) ( non empty strict ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) . (p : ( ( ) ( ) VectSpStr over L : ( ( ) ( ) 1-sorted ) ) ,n : ( ( ) ( ) BiModStr over L : ( ( ) ( ) 1-sorted ) ,p : ( ( ) ( ) VectSpStr over L : ( ( ) ( ) 1-sorted ) ) ) ) : ( ( ) ( ) set ) ;

end;

registration

cluster p : ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty right_complementable Abelian add-associative right_zeroed commutative right_unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable Abelian add-associative right_zeroed commutative right-distributive left-distributive right_unital distributive V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of L : ( ( non empty right_complementable Abelian add-associative right_zeroed commutative right_unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable Abelian add-associative right_zeroed commutative right-distributive left-distributive right_unital distributive V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) `^ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty right_complementable Abelian add-associative right_zeroed commutative right_unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable Abelian add-associative right_zeroed commutative right-distributive left-distributive right_unital distributive V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) -> Function-like quasi_total finite-Support ;

end;

theorem :: POLYNOM5:15

theorem :: POLYNOM5:16

theorem :: POLYNOM5:17

theorem :: POLYNOM5:18

theorem :: POLYNOM5:19

theorem :: POLYNOM5:20

theorem :: POLYNOM5:21

theorem :: POLYNOM5:22

theorem :: POLYNOM5:23

definition

let L be ( ( non empty ) ( non empty ) multMagma ) ;
let p be ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ;
let v be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

func v * p -> ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( ) set ) ) means :: POLYNOM5:def 3
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) holds it : ( ( ) ( ) Element of p : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) = v : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) * (p : ( ( non empty ) ( non empty ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ;

end;

registration

let L be ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed right-distributive V174() V175() V176() V177() ) doubleLoopStr ) ;
let p be ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed right-distributive V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Polynomial of L : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed right-distributive V174() V175() V176() V177() ) doubleLoopStr ) ) ;
let v be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

cluster v : ( ( ) ( ) Element of the carrier of L : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed right-distributive V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) * p : ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed right-distributive V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of L : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed right-distributive V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty right_complementable add-associative right_zeroed right-distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed right-distributive V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) -> Function-like quasi_total finite-Support ;

end;

theorem :: POLYNOM5:24

theorem :: POLYNOM5:25

theorem :: POLYNOM5:26

theorem :: POLYNOM5:27

for L being ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) multLoopStr )
for p being ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) holds (1. L : ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) multLoopStr ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) multLoopStr ) : ( ( ) ( non empty ) set ) ) * p : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) = p : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) multLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ;

theorem :: POLYNOM5:28

theorem :: POLYNOM5:29

theorem :: POLYNOM5:30

theorem :: POLYNOM5:31

definition

let L be ( ( non empty ) ( non empty ) ZeroStr ) ;
let z0, z1 be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

func <%z0,z1%> -> ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( ) set ) ) equals :: POLYNOM5:def 4
((0_. L : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like non empty total quasi_total ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) +* (0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ,z0 : ( ( non empty ) ( non empty ) set ) )) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) +* (1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ,z1 : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

end;

theorem :: POLYNOM5:32

for L being ( ( non empty ) ( non empty ) ZeroStr )
for z0 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( <%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) = z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) >= 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) holds
<%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) = 0. L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( zero ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: POLYNOM5:33

for L being ( ( non empty ) ( non empty ) ZeroStr )
for z0 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> 0. L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( zero ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) holds
len <%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) = 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ;

theorem :: POLYNOM5:34

for L being ( ( non empty ) ( non empty ) ZeroStr ) holds <%(0. L : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( zero ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) = 0_. L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) ;

theorem :: POLYNOM5:35

theorem :: POLYNOM5:36

theorem :: POLYNOM5:37

for L being ( ( non empty right_complementable add-associative right_zeroed unital ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed unital V174() V175() V176() V177() ) doubleLoopStr )
for z0, x being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds eval (<%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty right_complementable add-associative right_zeroed unital ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed unital V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of b₁ : ( ( non empty right_complementable add-associative right_zeroed unital ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed unital V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) ,x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty right_complementable add-associative right_zeroed unital ) ( non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed unital V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: POLYNOM5:38

for L being ( ( non empty ) ( non empty ) ZeroStr )
for z0, z1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( <%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) = z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & <%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) . 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) = z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() ) Nat) >= 2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) holds
<%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() ) Nat) : ( ( ) ( ) set ) = 0. L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( zero ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) ) ) ;

registration

let L be ( ( non empty ) ( non empty ) ZeroStr ) ;
let z0, z1 be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

cluster <%z0 : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) ,z1 : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) -> Function-like quasi_total finite-Support ;

end;

theorem :: POLYNOM5:39

for L being ( ( non empty ) ( non empty ) ZeroStr )
for z0, z1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds len <%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) <= 2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ;

theorem :: POLYNOM5:40

for L being ( ( non empty ) ( non empty ) ZeroStr )
for z0, z1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> 0. L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( zero ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) holds
len <%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,z1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) = 2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ;

theorem :: POLYNOM5:41

for L being ( ( non empty ) ( non empty ) ZeroStr )
for z0 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> 0. L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( zero ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) holds
len <%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,(0. L : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( zero ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) = 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ;

theorem :: POLYNOM5:42

for L being ( ( non empty ) ( non empty ) ZeroStr ) holds <%(0. L : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( zero ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) ,(0. L : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( zero ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) sequence of ( ( ) ( non empty ) set ) ) = 0_. L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) ;

theorem :: POLYNOM5:43

for L being ( ( non empty ) ( non empty ) ZeroStr )
for z0 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds <%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,(0. L : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( zero ) Element of the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) sequence of ( ( ) ( non empty ) set ) ) = <%z0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) %> : ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of b₁ : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) ;

theorem :: POLYNOM5:44

theorem :: POLYNOM5:45

theorem :: POLYNOM5:46

theorem :: POLYNOM5:47

theorem :: POLYNOM5:48

begin

definition

let L be ( ( non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable Abelian add-associative right_zeroed unital commutative right-distributive left-distributive right_unital well-unital distributive left_unital V174() V175() V176() V177() ) doubleLoopStr ) ;
let p, q be ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable Abelian add-associative right_zeroed unital commutative right-distributive left-distributive right_unital well-unital distributive left_unital V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Polynomial of L : ( ( non empty right_complementable Abelian add-associative right_zeroed commutative well-unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable Abelian add-associative right_zeroed unital commutative right-distributive left-distributive right_unital well-unital distributive left_unital V174() V175() V176() V177() ) doubleLoopStr ) ) ;

end;

theorem :: POLYNOM5:49

theorem :: POLYNOM5:50

theorem :: POLYNOM5:51

theorem :: POLYNOM5:52

theorem :: POLYNOM5:53

begin

definition

let L be ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) ;
let p be ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Polynomial of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) ) ;
let x be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

pred x is_a_root_of p means :: POLYNOM5:def 6
eval (p : ( ( non empty ) ( non empty ) set ) ,x : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) = 0. L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ;

end;

definition

attr p is with_roots means :: POLYNOM5:def 7
ex x being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_a_root_of p : ( ( non empty ) ( non empty ) set ) ;

end;

theorem :: POLYNOM5:54

for L being ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) holds 0_. L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) is with_roots ;

registration

let L be ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) ;

cluster 0_. L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) -> Function-like quasi_total with_roots ;

end;

theorem :: POLYNOM5:55

for L being ( ( non empty unital ) ( non empty unital ) doubleLoopStr )
for x being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_a_root_of 0_. L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support with_roots ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) ;

registration

let L be ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) ;

cluster Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support with_roots for ( ( ) ( ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) ;

end;

definition

let L be ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) ;

end;

definition

func Roots p -> ( ( ) ( ) Subset of ) means :: POLYNOM5:def 9
for x being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in it : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) iff x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_a_root_of p : ( ( non empty ) ( non empty ) set ) );

end;

definition

let L be ( ( non empty almost_left_invertible associative commutative well-unital distributive ) ( non empty almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ;
let p be ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty almost_left_invertible associative commutative well-unital distributive ) ( non empty almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Polynomial of L : ( ( non empty almost_left_invertible associative commutative well-unital distributive ) ( non empty almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital ) doubleLoopStr ) ) ;

func NormPolynomial p -> ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) means :: POLYNOM5:def 10
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) holds it : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = (p : ( ( non empty ) ( non empty ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) / (p : ( ( non empty ) ( non empty ) set ) . ((len p : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) -' 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ;

end;

registration

let L be ( ( non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable almost_left_invertible add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) ;
let p be ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable almost_left_invertible add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Polynomial of L : ( ( non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable almost_left_invertible add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) ) ;

cluster NormPolynomial p : ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable almost_left_invertible add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Element of K27(K28(NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) , the carrier of L : ( ( non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable almost_left_invertible add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive ) ( non empty left_add-cancelable right_add-cancelable right_complementable almost_left_invertible add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) -> Function-like quasi_total finite-Support ;

end;

theorem :: POLYNOM5:56

theorem :: POLYNOM5:57

errorfrm ;

theorem :: POLYNOM5:58

errorfrm ;

theorem :: POLYNOM5:59

errorfrm ;

theorem :: POLYNOM5:60

errorfrm ;

theorem :: POLYNOM5:61

errorfrm ;

theorem :: POLYNOM5:62

id COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) : ( ( total ) ( Relation-like COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -defined COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -valued Function-like one-to-one non empty total quasi_total complex-valued ) Element of K27(K28(COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ,COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) : ( ( ) ( V24() complex-valued ) set ) ) : ( ( ) ( V24() ) set ) ) is_continuous_on COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ;

theorem :: POLYNOM5:63

for x being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) holds COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) --> x : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -defined COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -valued Function-like constant non empty total quasi_total complex-valued ) Element of K27(K28(COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ,COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) : ( ( ) ( V24() complex-valued ) set ) ) : ( ( ) ( V24() ) set ) ) is_continuous_on COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ;

definition

let L be ( ( non empty unital ) ( non empty unital ) multMagma ) ;
let x be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ;

func FPower (x,n) -> ( ( Function-like quasi_total ) ( Relation-like the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) means :: POLYNOM5:def 11
for y being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds it : ( ( ) ( ) Element of x : ( ( non empty ) ( non empty ) set ) ) . y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = x : ( ( non empty ) ( non empty ) set ) * ((power L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) ) : ( ( Function-like quasi_total ) ( Relation-like K28( the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ,NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V24() V56() V57() V58() V59() V62() ) set ) -valued INT : ( ( ) ( non empty V24() V56() V57() V58() V59() V60() V62() ) set ) -valued V24() complex-valued ext-real-valued real-valued natural-valued ) set ) -defined the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K27(K28(K28( the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ,NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( ) ( RAT : ( ( ) ( non empty V24() V56() V57() V58() V59() V62() ) set ) -valued INT : ( ( ) ( non empty V24() V56() V57() V58() V59() V60() V62() ) set ) -valued V24() complex-valued ext-real-valued real-valued natural-valued ) set ) , the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( ) ( V24() ) set ) ) . (y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,n : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ;

end;

theorem :: POLYNOM5:64

for L being ( ( non empty unital ) ( non empty unital ) multMagma ) holds FPower ((1_ L : ( ( non empty unital ) ( non empty unital ) multMagma ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) ) ,1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = id the carrier of L : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) : ( ( total ) ( Relation-like the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one non empty total quasi_total ) Element of K27(K28( the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) , the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: POLYNOM5:65

theorem :: POLYNOM5:66

for L being ( ( non empty unital ) ( non empty unital ) multMagma )
for x being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds FPower (x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() V29() complex real ext-real non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = the carrier of L : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) --> x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like constant non empty total quasi_total ) Element of K27(K28( the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) , the carrier of b₁ : ( ( non empty unital ) ( non empty unital ) multMagma ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: POLYNOM5:67

theorem :: POLYNOM5:68

for x being ( ( ) ( complex ) Element of ( ( ) ( non empty V19() ) set ) ) ex x1 being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) st
( x : ( ( ) ( complex ) Element of ( ( ) ( non empty V19() ) set ) ) = x1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) & FPower (x : ( ( ) ( complex ) Element of ( ( ) ( non empty V19() ) set ) ) ,2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty V24() V29() complex real ext-real positive non negative V54() V55() V56() V57() V58() V59() V60() V61() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) -defined the carrier of F_Complex : ( ( strict ) ( non empty non degenerated V94() left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital domRing-like V174() V175() V176() V177() ) doubleLoopStr ) : ( ( ) ( non empty V19() ) set ) -valued Function-like non empty total quasi_total ) Function of ( ( ) ( non empty V19() ) set ) , ( ( ) ( non empty V19() ) set ) ) = x1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) (#) ((id COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) : ( ( total ) ( Relation-like COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -defined COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -valued Function-like one-to-one non empty total quasi_total complex-valued ) Element of K27(K28(COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ,COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) : ( ( ) ( V24() complex-valued ) set ) ) : ( ( ) ( V24() ) set ) ) (#) (id COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) : ( ( total ) ( Relation-like COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -defined COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -valued Function-like one-to-one non empty total quasi_total complex-valued ) Element of K27(K28(COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ,COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) : ( ( ) ( V24() complex-valued ) set ) ) : ( ( ) ( V24() ) set ) ) ) : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -defined COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -valued Function-like non empty total quasi_total complex-valued ) Element of K27(K28(COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ,COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) : ( ( ) ( V24() complex-valued ) set ) ) : ( ( ) ( V24() ) set ) ) : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -defined COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) -valued Function-like non empty total quasi_total complex-valued ) Element of K27(K28(COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ,COMPLEX : ( ( ) ( non empty V24() V56() V62() ) set ) ) : ( ( ) ( V24() complex-valued ) set ) ) : ( ( ) ( V24() ) set ) ) ) ;

theorem :: POLYNOM5:69

theorem :: POLYNOM5:70

definition

let L be ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) doubleLoopStr ) ;
let p be ( ( Function-like quasi_total finite-Support ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty V24() V29() V30() V56() V57() V58() V59() V60() V61() V62() ) Element of K27(REAL : ( ( ) ( non empty V24() V56() V57() V58() V62() ) set ) ) : ( ( ) ( V24() ) set ) ) -defined the carrier of L : ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total finite-Support ) Polynomial of L : ( ( non empty well-unital ) ( non empty unital right_unital well-unital left_unital ) doubleLoopStr ) ) ;

func Polynomial-Function (L,p) -> ( ( Function-like quasi_total ) ( Relation-like the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -defined the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) means :: POLYNOM5:def 12
for x being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds it : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) = eval (p : ( ( non empty ) ( non empty ) set ) ,x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty unital ) ( non empty unital ) doubleLoopStr ) : ( ( ) ( non empty ) set ) ) ;

end;

theorem :: POLYNOM5:71

theorem :: POLYNOM5:72

theorem :: POLYNOM5:73

theorem :: POLYNOM5:74

registration 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 for ( ( ) ( ) doubleLoopStr ) ;

end;