:: POLYEQ_3 semantic presentation

begin

definition
let z be ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;
:: original: ^2
redefine func z ^2 -> ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) equals :: POLYEQ_3:def 1
(((Re z : ( ( complex ) ( complex ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((Im z : ( ( complex ) ( complex ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((Re z : ( ( complex ) ( complex ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * (Im z : ( ( complex ) ( complex ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;
end;

definition
let a, b, c be ( ( ) ( complex real ext-real ) Real) ;
let z be ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;
:: original: Polynom
redefine func Polynom (a,b,c,z) -> ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;
end;

theorem :: POLYEQ_3:1
for a, b, c being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st a : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) >= 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((- b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((- b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - (b : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ;

theorem :: POLYEQ_3:2
for a, b, c being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st a : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) < 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (b : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((sqrt (- (delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (b : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- ((sqrt (- (delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:3
for b, c being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st b : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds Polynom (0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - (c : ( ( ) ( complex real ext-real ) Real) / b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ;

theorem :: POLYEQ_3:4
for a, b, c being ( ( ) ( complex real ext-real ) Real)
for z, z1, z2 being ( ( complex ) ( complex ) number ) st a : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for z being ( ( complex ) ( complex ) number ) holds Polynom (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) = Quard (a : ( ( ) ( complex real ext-real ) Real) ,z1 : ( ( complex ) ( complex ) number ) ,z2 : ( ( complex ) ( complex ) number ) ,z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( ) set ) ) holds
( b : ( ( ) ( complex real ext-real ) Real) / a : ( ( ) ( complex real ext-real ) Real) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = - (z1 : ( ( complex ) ( complex ) number ) + z2 : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & c : ( ( ) ( complex real ext-real ) Real) / a : ( ( ) ( complex real ext-real ) Real) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = z1 : ( ( complex ) ( complex ) number ) * z2 : ( ( complex ) ( complex ) number ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) ;

definition
let z be ( ( complex ) ( complex ) number ) ;
func z ^3 -> ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) equals :: POLYEQ_3:def 2
(z : ( ( complex ) ( complex ) set ) ^2) : ( ( ) ( complex ) set ) * z : ( ( complex ) ( complex ) set ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;
end;

definition
let a, b, c, d, z be ( ( complex ) ( complex ) number ) ;
redefine func Polynom (a,b,c,d,z) equals :: POLYEQ_3:def 3
(((a : ( ( complex ) ( complex ) set ) * (z : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^3) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) + (b : ( ( ) ( ) set ) * (z : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) + (c : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * z : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) + d : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;
end;

theorem :: POLYEQ_3:5
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
( Re (z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^3) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = ((Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * ((Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & Im (z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^3) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = (- ((Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * (Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) ;

theorem :: POLYEQ_3:6
for a, b, c, d, a9, b9, c9, d9 being ( ( ) ( complex real ext-real ) Real) st ( for z being ( ( complex ) ( complex ) number ) holds Polynom (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) ,z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) = Polynom (a9 : ( ( ) ( complex real ext-real ) Real) ,b9 : ( ( ) ( complex real ext-real ) Real) ,c9 : ( ( ) ( complex real ext-real ) Real) ,d9 : ( ( ) ( complex real ext-real ) Real) ,z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) set ) ) holds
( a : ( ( ) ( complex real ext-real ) Real) = a9 : ( ( ) ( complex real ext-real ) Real) & b : ( ( ) ( complex real ext-real ) Real) = b9 : ( ( ) ( complex real ext-real ) Real) & c : ( ( ) ( complex real ext-real ) Real) = c9 : ( ( ) ( complex real ext-real ) Real) & d : ( ( ) ( complex real ext-real ) Real) = d9 : ( ( ) ( complex real ext-real ) Real) ) ;

theorem :: POLYEQ_3:7
for b, c, d being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st b : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & delta (b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) >= 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((- c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (delta (b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((- c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (delta (b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - (c : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ;

theorem :: POLYEQ_3:8
for b, c, d being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st b : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & delta (b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) < 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (c : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((sqrt (- (delta (b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (c : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- ((sqrt (- (delta (b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:9
for a, c being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st a : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & (4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * c : ( ( ) ( complex real ext-real ) Real) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) <= 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (a : ( ( ) ( complex real ext-real ) Real) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,c : ( ( ) ( complex real ext-real ) Real) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (sqrt (- ((4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (sqrt (- ((4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: POLYEQ_3:10
for a, b, c being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st a : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) >= 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((- b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((- b : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - (b : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: POLYEQ_3:11
for a, b, c being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st a : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) < 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (b : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((sqrt (- (delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (b : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- ((sqrt (- (delta (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: POLYEQ_3:12
for y, a being ( ( ) ( complex real ext-real ) Real) holds
( not y : ( ( ) ( complex real ext-real ) Real) ^2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = a : ( ( ) ( complex real ext-real ) Real) or y : ( ( ) ( complex real ext-real ) Real) = sqrt a : ( ( ) ( complex real ext-real ) Real) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) or y : ( ( ) ( complex real ext-real ) Real) = - (sqrt a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) ;

theorem :: POLYEQ_3:13
for a, c, d being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st a : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (a : ( ( ) ( complex real ext-real ) Real) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for u, v being ( ( ) ( complex real ext-real ) Real) st Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = u : ( ( ) ( complex real ext-real ) Real) + v : ( ( ) ( complex real ext-real ) Real) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * v : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * u : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (c : ( ( ) ( complex real ext-real ) Real) / a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((- (d : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((- (d : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((- (d : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((- (d : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((- (d : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((- (d : ( ( ) ( complex real ext-real ) Real) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ;

theorem :: POLYEQ_3:14
for a, c, d being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st a : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (a : ( ( ) ( complex real ext-real ) Real) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for u, v being ( ( ) ( complex real ext-real ) Real) st Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = u : ( ( ) ( complex real ext-real ) Real) + v : ( ( ) ( complex real ext-real ) Real) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * v : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * u : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (c : ( ( ) ( complex real ext-real ) Real) / (4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (c : ( ( ) ( complex real ext-real ) Real) / a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((sqrt ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (c : ( ( ) ( complex real ext-real ) Real) / a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (c : ( ( ) ( complex real ext-real ) Real) / a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((sqrt ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (c : ( ( ) ( complex real ext-real ) Real) / a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (c : ( ( ) ( complex real ext-real ) Real) / a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((sqrt ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((d : ( ( ) ( complex real ext-real ) Real) / (16 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((c : ( ( ) ( complex real ext-real ) Real) / (12 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (c : ( ( ) ( complex real ext-real ) Real) / a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:15
for a, b, c, d being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st a : ( ( ) ( complex real ext-real ) Real) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (a : ( ( ) ( complex real ext-real ) Real) ,b : ( ( ) ( complex real ext-real ) Real) ,c : ( ( ) ( complex real ext-real ) Real) ,d : ( ( ) ( complex real ext-real ) Real) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for u, v, x1 being ( ( ) ( complex real ext-real ) Real) st x1 : ( ( ) ( complex real ext-real ) Real) = (Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = (u : ( ( ) ( complex real ext-real ) Real) + v : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) & ((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * u : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * v : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root (((- ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (6 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (9 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root (((- ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (6 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (9 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root (((- ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (6 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (9 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root (((- ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (6 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (9 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root (((- ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (6 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (9 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root (((- ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (6 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (sqrt (((((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * ((b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * d : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((((3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / (9 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) - (b : ( ( ) ( complex real ext-real ) Real) / (3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:16
for z1, z2, z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - (z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

begin

theorem :: POLYEQ_3:17
for z1, z2, z3, s1, s2, s3 being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st ( for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds Polynom (z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = Polynom (s1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,s2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,s3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) ) holds
( z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = s1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = s2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & z3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = s3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) ;

theorem :: POLYEQ_3:18
for a, b being ( ( ) ( complex real ext-real ) Real) holds
( ((- a : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt ((a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (b : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) >= 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & (a : ( ( ) ( complex real ext-real ) Real) + (sqrt ((a : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (b : ( ( ) ( complex real ext-real ) Real) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) >= 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: POLYEQ_3:19
for z, s being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) >= 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:20
for z, s being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) > 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = sqrt (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - (sqrt (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ;

theorem :: POLYEQ_3:21
for z, s being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) < 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (sqrt (- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - ((sqrt (- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:22
for z, s being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) < 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:23
for z, s being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
( not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) ;

theorem :: POLYEQ_3:24
for z1, z2, z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - (z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: POLYEQ_3:25
for z1, z3, z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for s being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
( not s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - (z3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) ;

theorem :: POLYEQ_3:26
for z1, z2, z3, z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for h, t being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^2) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) - (z3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & t : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) - t : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((- (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) - t : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) - t : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((- (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) - t : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:27
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
( z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) set ) = (z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) * z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) * z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) set ) = (z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^2) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) * z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) |^ 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) set ) = z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) ;

theorem :: POLYEQ_3:28
for z1, z2, z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - (z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: POLYEQ_3:29
for z1, z3, z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for s being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
( not s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = - (z3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) or z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = (- (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) ;

theorem :: POLYEQ_3:30
for z1, z2, z3, z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & Polynom (z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,z3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ,0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) set ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for s, h, t being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) st h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ^2) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) - (z3 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & t : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = z2 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * z1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) - t : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((- (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) - t : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) & not z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((- (sqrt (((- (Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) - t : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) holds
z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) = ((- (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sqrt (((- (Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (sqrt (((Re h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((Im h : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) - t : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:31
for x being ( ( real ) ( complex real ext-real ) number )
for n being ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds ((cos x : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) + ((sin x : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) |^ n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) set ) = (cos (n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sin (n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:32
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) )
for n being ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) |^ n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) set ) = ((|.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) to_power n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( real ) ( complex real ext-real ) set ) * (cos (n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (Arg z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((|.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) to_power n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( real ) ( complex real ext-real ) set ) * (sin (n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * (Arg z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:33
for n, k being ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( complex real ext-real ) Real) st n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((cos ((x : ( ( ) ( complex real ext-real ) Real) + ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * k : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sin ((x : ( ( ) ( complex real ext-real ) Real) + ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * k : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) |^ n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) set ) = (cos x : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sin x : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ;

theorem :: POLYEQ_3:34
for z being ( ( complex ) ( complex ) number )
for n, k being ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
z : ( ( complex ) ( complex ) number ) = (((n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root |.z : ( ( complex ) ( complex ) number ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * (cos (((Arg z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * k : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root |.z : ( ( complex ) ( complex ) number ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * (sin (((Arg z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * k : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) |^ n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) set ) ;

theorem :: POLYEQ_3:35
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) )
for n being ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) )
for k being ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds ((n : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root |.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * (cos (((Arg z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * k : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / n : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + (((n : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) -root |.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * (sin (((Arg z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * k : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / n : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) is ( ( ) ( complex ) CRoot of n : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) ;

theorem :: POLYEQ_3:36
for z being ( ( complex ) ( complex ) number )
for v being ( ( ) ( complex ) CRoot of 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( complex ) ( complex ) number ) ) holds v : ( ( ) ( complex ) CRoot of 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,b1 : ( ( complex ) ( complex ) number ) ) = z : ( ( complex ) ( complex ) number ) ;

theorem :: POLYEQ_3:37
for n being ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative ) Nat)
for v being ( ( ) ( complex ) CRoot of n : ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative ) Nat) , 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds v : ( ( ) ( complex ) CRoot of b1 : ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative ) Nat) , 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: POLYEQ_3:38
for n being ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) )
for z being ( ( complex ) ( complex ) number )
for v being ( ( ) ( complex ) CRoot of n : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( complex ) ( complex ) number ) ) st v : ( ( ) ( complex ) CRoot of b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,b2 : ( ( complex ) ( complex ) number ) ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
z : ( ( complex ) ( complex ) number ) = 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: POLYEQ_3:39
for n being ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) )
for k being ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds (cos (((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * k : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / n : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) + ((sin (((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * k : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) / n : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) is ( ( ) ( complex ) CRoot of n : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: POLYEQ_3:40
for z, s being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) )
for n being ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) st s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural complex real ext-real non positive non negative V43() V44() V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) |^ n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) set ) = z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) |^ n : ( ( ) ( ordinal natural complex real ext-real non negative V43() V44() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( V69() V70() V71() V72() V73() V74() V75() ) Element of K19(REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) set ) holds
|.s : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) = |.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( V69() V75() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( V69() V70() V71() V75() ) set ) ) ;