:: POLYEQ_3 semantic presentation

Lm1: 0 = [*0,0*]
by ARYTM_0:def 7;

definition
let a be Real;
let z be Element of COMPLEX ;
:: original: *
redefine func a * z -> Element of COMPLEX ;
correctness
coherence
a * z is Element of COMPLEX
;
by XCMPLX_0:def 2;
:: original: +
redefine func a + z -> Element of COMPLEX ;
correctness
coherence
a + z is Element of COMPLEX
;
by XCMPLX_0:def 2;
end;

definition
let z be Element of COMPLEX ;
:: original: ^2
redefine func z ^2 -> Element of COMPLEX equals :: POLYEQ_3:def 1
(((Re z) ^2 ) - ((Im z) ^2 )) + ((2 * ((Re z) * (Im z))) * <i> );
compatibility
for b1 being Element of COMPLEX holds
( b1 = z ^2 iff b1 = (((Re z) ^2 ) - ((Im z) ^2 )) + ((2 * ((Re z) * (Im z))) * <i> ) )
proof end;
correctness
coherence
z ^2 is Element of COMPLEX
;
by XCMPLX_0:def 2;
end;

:: deftheorem defines ^2 POLYEQ_3:def 1 :
for z being Element of COMPLEX holds z ^2 = (((Re z) ^2 ) - ((Im z) ^2 )) + ((2 * ((Re z) * (Im z))) * <i> );

definition
let a, b, c be Real;
let z be Element of COMPLEX ;
:: original: Polynom
redefine func Polynom a,b,c,z -> Element of COMPLEX ;
coherence
Polynom a,b,c,z is Element of COMPLEX
by XCMPLX_0:def 2;
end;

theorem :: POLYEQ_3:1
for a, c, b, d being Real holds (a + (c * <i> )) * (b + (d * <i> )) = ((a * b) - (c * d)) + (((a * d) + (b * c)) * <i> ) ;

theorem :: POLYEQ_3:2
for x, y being Real holds (x + (y * <i> )) ^2 = ((x ^2 ) - (y ^2 )) + (((2 * x) * y) * <i> ) ;

theorem :: POLYEQ_3:3
canceled;

theorem Th4: :: POLYEQ_3:4
for a, b, c being Real
for z being Element of COMPLEX st a <> 0 & delta a,b,c >= 0 & Polynom a,b,c,z = 0 & not z = ((- b) + (sqrt (delta a,b,c))) / (2 * a) & not z = ((- b) - (sqrt (delta a,b,c))) / (2 * a) holds
z = - (b / (2 * a))
proof end;

theorem Th5: :: POLYEQ_3:5
for a, b, c being Real
for z being Element of COMPLEX st a <> 0 & delta a,b,c < 0 & Polynom a,b,c,z = 0 & not z = (- (b / (2 * a))) + (((sqrt (- (delta a,b,c))) / (2 * a)) * <i> ) holds
z = (- (b / (2 * a))) + ((- ((sqrt (- (delta a,b,c))) / (2 * a))) * <i> )
proof end;

theorem :: POLYEQ_3:6
for b, c being Real
for z being Element of COMPLEX st b <> 0 & ( for z being Element of COMPLEX holds Polynom 0,b,c,z = 0 ) holds
z = - (c / b)
proof end;

theorem :: POLYEQ_3:7
for a, b, c being Real
for z, z1, z2 being complex number st a <> 0 & ( for z being complex number holds Polynom a,b,c,z = Quard a,z1,z2,z ) holds
( b / a = - (z1 + z2) & c / a = z1 * z2 )
proof end;

definition
let z be complex number ;
func z ^3 -> Element of COMPLEX equals :: POLYEQ_3:def 2
(z ^2 ) * z;
correctness
coherence
(z ^2 ) * z is Element of COMPLEX
;
by XCMPLX_0:def 2;
end;

:: deftheorem defines ^3 POLYEQ_3:def 2 :
for z being complex number holds z ^3 = (z ^2 ) * z;

Lm2: for z being complex number holds z |^ 2 = z * z
proof end;

definition
let a, b, c, d, z be complex number ;
redefine func Polynom a,b,c,d,z equals :: POLYEQ_3:def 3
(((a * (z ^3 )) + (b * (z ^2 ))) + (c * z)) + d;
compatibility
for b1 being set holds
( b1 = Polynom a,b,c,d,z iff b1 = (((a * (z ^3 )) + (b * (z ^2 ))) + (c * z)) + d )
proof end;
end;

:: deftheorem defines Polynom POLYEQ_3:def 3 :
for a, b, c, d, z being complex number holds Polynom a,b,c,d,z = (((a * (z ^3 )) + (b * (z ^2 ))) + (c * z)) + d;

theorem :: POLYEQ_3:8
0 ^3 = 0 ;

theorem :: POLYEQ_3:9
1 ^3 = 1 ;

theorem :: POLYEQ_3:10
(- 1) ^3 = - 1 ;

theorem Th11: :: POLYEQ_3:11
for z being Element of COMPLEX holds
( Re (z ^3 ) = ((Re z) |^ 3) - ((3 * (Re z)) * ((Im z) ^2 )) & Im (z ^3 ) = (- ((Im z) |^ 3)) + ((3 * ((Re z) ^2 )) * (Im z)) )
proof end;

theorem :: POLYEQ_3:12
for a, b, c, d, a', b', c', d' being Real st ( for z being complex number holds Polynom a,b,c,d,z = Polynom a',b',c',d',z ) holds
( a = a' & b = b' & c = c' & d = d' )
proof end;

theorem :: POLYEQ_3:13
for z, h being Element of COMPLEX holds (z + h) ^3 = (((z ^3 ) + ((3 * h) * (z ^2 ))) + ((3 * (h ^2 )) * z)) + (h ^3 ) ;

theorem :: POLYEQ_3:14
for z, h being Element of COMPLEX holds (z * h) ^3 = (z ^3 ) * (h ^3 ) ;

theorem :: POLYEQ_3:15
for b, c, d being Real
for z being Element of COMPLEX st b <> 0 & delta b,c,d >= 0 & Polynom 0,b,c,d,z = 0 & not z = ((- c) + (sqrt (delta b,c,d))) / (2 * b) & not z = ((- c) - (sqrt (delta b,c,d))) / (2 * b) holds
z = - (c / (2 * b))
proof end;

theorem :: POLYEQ_3:16
for b, c, d being Real
for z being Element of COMPLEX st b <> 0 & delta b,c,d < 0 & Polynom 0,b,c,d,z = 0 & not z = (- (c / (2 * b))) + (((sqrt (- (delta b,c,d))) / (2 * b)) * <i> ) holds
z = (- (c / (2 * b))) + ((- ((sqrt (- (delta b,c,d))) / (2 * b))) * <i> )
proof end;

theorem :: POLYEQ_3:17
for a, c being Real
for z being Element of COMPLEX st a <> 0 & (4 * a) * c <= 0 & Polynom a,0,c,0,z = 0 & not z = (sqrt (- ((4 * a) * c))) / (2 * a) & not z = (- (sqrt (- ((4 * a) * c)))) / (2 * a) holds
z = 0
proof end;

theorem :: POLYEQ_3:18
for a, b, c being Real
for z being Element of COMPLEX st a <> 0 & delta a,b,c >= 0 & Polynom a,b,c,0,z = 0 & not z = ((- b) + (sqrt (delta a,b,c))) / (2 * a) & not z = ((- b) - (sqrt (delta a,b,c))) / (2 * a) & not z = - (b / (2 * a)) holds
z = 0
proof end;

theorem :: POLYEQ_3:19
for a, b, c being Real
for z being Element of COMPLEX st a <> 0 & delta a,b,c < 0 & Polynom a,b,c,0,z = 0 & not z = (- (b / (2 * a))) + (((sqrt (- (delta a,b,c))) / (2 * a)) * <i> ) & not z = (- (b / (2 * a))) + ((- ((sqrt (- (delta a,b,c))) / (2 * a))) * <i> ) holds
z = 0
proof end;

theorem Th20: :: POLYEQ_3:20
for y, a being Real holds
( not y ^2 = a or y = sqrt a or y = - (sqrt a) )
proof end;

theorem :: POLYEQ_3:21
for a, c, d being Real
for z being Element of COMPLEX st a <> 0 & Im z = 0 & Polynom a,0,c,d,z = 0 holds
for u, v being Real st Re z = u + v & ((3 * v) * u) + (c / a) = 0 & not z = (3 -root ((- (d / (2 * a))) + (sqrt (((d ^2 ) / (4 * (a ^2 ))) + ((c / (3 * a)) |^ 3))))) + (3 -root ((- (d / (2 * a))) - (sqrt (((d ^2 ) / (4 * (a ^2 ))) + ((c / (3 * a)) |^ 3))))) & not z = (3 -root ((- (d / (2 * a))) + (sqrt (((d ^2 ) / (4 * (a ^2 ))) + ((c / (3 * a)) |^ 3))))) + (3 -root ((- (d / (2 * a))) + (sqrt (((d ^2 ) / (4 * (a ^2 ))) + ((c / (3 * a)) |^ 3))))) holds
z = (3 -root ((- (d / (2 * a))) - (sqrt (((d ^2 ) / (4 * (a ^2 ))) + ((c / (3 * a)) |^ 3))))) + (3 -root ((- (d / (2 * a))) - (sqrt (((d ^2 ) / (4 * (a ^2 ))) + ((c / (3 * a)) |^ 3)))))
proof end;

theorem :: POLYEQ_3:22
for a, c, d being Real
for z being Element of COMPLEX st a <> 0 & Im z <> 0 & Polynom a,0,c,d,z = 0 holds
for u, v being Real st Re z = u + v & ((3 * v) * u) + (c / (4 * a)) = 0 & c / a >= 0 & not z = ((3 -root ((d / (16 * a)) + (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3))))) + (3 -root ((d / (16 * a)) - (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) + ((sqrt ((3 * (((3 -root ((d / (16 * a)) + (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3))))) + (3 -root ((d / (16 * a)) - (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) ^2 )) + (c / a))) * <i> ) & not z = ((3 -root ((d / (16 * a)) + (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3))))) + (3 -root ((d / (16 * a)) - (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) - ((sqrt ((3 * (((3 -root ((d / (16 * a)) + (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3))))) + (3 -root ((d / (16 * a)) - (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) ^2 )) + (c / a))) * <i> ) & not z = (2 * (3 -root ((d / (16 * a)) + (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) + ((sqrt ((3 * ((2 * (3 -root ((d / (16 * a)) + (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) ^2 )) + (c / a))) * <i> ) & not z = (2 * (3 -root ((d / (16 * a)) + (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) - ((sqrt ((3 * ((2 * (3 -root ((d / (16 * a)) + (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) ^2 )) + (c / a))) * <i> ) & not z = (2 * (3 -root ((d / (16 * a)) - (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) + ((sqrt ((3 * ((2 * (3 -root ((d / (16 * a)) - (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) ^2 )) + (c / a))) * <i> ) holds
z = (2 * (3 -root ((d / (16 * a)) - (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) - ((sqrt ((3 * ((2 * (3 -root ((d / (16 * a)) - (sqrt (((d / (16 * a)) ^2 ) + ((c / (12 * a)) |^ 3)))))) ^2 )) + (c / a))) * <i> )
proof end;

theorem :: POLYEQ_3:23
for a, b, c, d being Real
for z being Element of COMPLEX st a <> 0 & Polynom a,b,c,d,z = 0 & Im z = 0 holds
for u, v, x1 being Real st x1 = (Re z) + (b / (3 * a)) & Re z = (u + v) - (b / (3 * a)) & ((3 * u) * v) + ((((3 * a) * c) - (b ^2 )) / (3 * (a ^2 ))) = 0 & not z = (((3 -root (((- ((b / (3 * a)) |^ 3)) - ((((3 * a) * d) - (b * c)) / (6 * (a ^2 )))) + (sqrt (((((2 * ((b / (3 * a)) |^ 3)) + ((((3 * a) * d) - (b * c)) / (3 * (a ^2 )))) ^2 ) / 4) + (((((3 * a) * c) - (b ^2 )) / (9 * (a ^2 ))) |^ 3))))) + (3 -root (((- ((b / (3 * a)) |^ 3)) - ((((3 * a) * d) - (b * c)) / (6 * (a ^2 )))) - (sqrt (((((2 * ((b / (3 * a)) |^ 3)) + ((((3 * a) * d) - (b * c)) / (3 * (a ^2 )))) ^2 ) / 4) + (((((3 * a) * c) - (b ^2 )) / (9 * (a ^2 ))) |^ 3)))))) - (b / (3 * a))) + (0 * <i> ) & not z = (((3 -root (((- ((b / (3 * a)) |^ 3)) - ((((3 * a) * d) - (b * c)) / (6 * (a ^2 )))) + (sqrt (((((2 * ((b / (3 * a)) |^ 3)) + ((((3 * a) * d) - (b * c)) / (3 * (a ^2 )))) ^2 ) / 4) + (((((3 * a) * c) - (b ^2 )) / (9 * (a ^2 ))) |^ 3))))) + (3 -root (((- ((b / (3 * a)) |^ 3)) - ((((3 * a) * d) - (b * c)) / (6 * (a ^2 )))) + (sqrt (((((2 * ((b / (3 * a)) |^ 3)) + ((((3 * a) * d) - (b * c)) / (3 * (a ^2 )))) ^2 ) / 4) + (((((3 * a) * c) - (b ^2 )) / (9 * (a ^2 ))) |^ 3)))))) - (b / (3 * a))) + (0 * <i> ) holds
z = (((3 -root (((- ((b / (3 * a)) |^ 3)) - ((((3 * a) * d) - (b * c)) / (6 * (a ^2 )))) - (sqrt (((((2 * ((b / (3 * a)) |^ 3)) + ((((3 * a) * d) - (b * c)) / (3 * (a ^2 )))) ^2 ) / 4) + (((((3 * a) * c) - (b ^2 )) / (9 * (a ^2 ))) |^ 3))))) + (3 -root (((- ((b / (3 * a)) |^ 3)) - ((((3 * a) * d) - (b * c)) / (6 * (a ^2 )))) - (sqrt (((((2 * ((b / (3 * a)) |^ 3)) + ((((3 * a) * d) - (b * c)) / (3 * (a ^2 )))) ^2 ) / 4) + (((((3 * a) * c) - (b ^2 )) / (9 * (a ^2 ))) |^ 3)))))) - (b / (3 * a))) + (0 * <i> )
proof end;

theorem Th24: :: POLYEQ_3:24
for z1, z2, z being Element of COMPLEX st z1 <> 0 & Polynom z1,z2,z = 0 holds
z = - (z2 / z1)
proof end;

theorem :: POLYEQ_3:25
for z2 being Element of COMPLEX st z2 <> 0 holds
for z being Element of COMPLEX holds not Polynom 0,z2,z = 0 ;

theorem :: POLYEQ_3:26
for z1, z2, z3, s1, s2, s3 being Element of COMPLEX st ( for z being Element of COMPLEX holds Polynom z1,z2,z3,z = Polynom s1,s2,s3,z ) holds
( z1 = s1 & z2 = s2 & z3 = s3 )
proof end;

theorem :: POLYEQ_3:27
for a, b being Real holds
( ((- a) + (sqrt ((a ^2 ) + (b ^2 )))) / 2 >= 0 & (a + (sqrt ((a ^2 ) + (b ^2 )))) / 2 >= 0 )
proof end;

theorem Th28: :: POLYEQ_3:28
for z, s being Element of COMPLEX st z ^2 = s & Im s >= 0 & not z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) holds
z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> )
proof end;

theorem :: POLYEQ_3:29
for z, s being Element of COMPLEX st z ^2 = s & Im s = 0 & Re s > 0 & not z = sqrt (Re s) holds
z = - (sqrt (Re s))
proof end;

theorem :: POLYEQ_3:30
for z, s being Element of COMPLEX st z ^2 = s & Im s = 0 & Re s < 0 & not z = (sqrt (- (Re s))) * <i> holds
z = - ((sqrt (- (Re s))) * <i> )
proof end;

theorem :: POLYEQ_3:31
for z, s being Element of COMPLEX st z ^2 = s & Im s < 0 & not z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) holds
z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> )
proof end;

theorem Th32: :: POLYEQ_3:32
for z, s being Element of COMPLEX holds
( not z ^2 = s or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) )
proof end;

theorem :: POLYEQ_3:33
for z being Element of COMPLEX holds Polynom 0,0,0,z = 0 ;

theorem Th34: :: POLYEQ_3:34
for z1, z being Element of COMPLEX st z1 <> 0 & Polynom z1,0,0,z = 0 holds
z = 0
proof end;

theorem :: POLYEQ_3:35
for z1, z2, z being Element of COMPLEX st z1 <> 0 & Polynom z1,z2,0,z = 0 & not z = - (z2 / z1) holds
z = 0
proof end;

theorem Th36: :: POLYEQ_3:36
for z1, z3, z being Element of COMPLEX st z1 <> 0 & Polynom z1,0,z3,z = 0 holds
for s being Element of COMPLEX holds
( not s = - (z3 / z1) or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) )
proof end;

theorem Th37: :: POLYEQ_3:37
for z1, z2, z3, z being Element of COMPLEX st z1 <> 0 & Polynom z1,z2,z3,z = 0 holds
for h, t being Element of COMPLEX st h = ((z2 / (2 * z1)) ^2 ) - (z3 / z1) & t = z2 / (2 * z1) & not z = ((sqrt (((Re h) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2)) + ((sqrt (((- (Re h)) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2)) * <i> )) - t & not z = ((- (sqrt (((Re h) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2))) + ((- (sqrt (((- (Re h)) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2))) * <i> )) - t & not z = ((sqrt (((Re h) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2)) + ((- (sqrt (((- (Re h)) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2))) * <i> )) - t holds
z = ((- (sqrt (((Re h) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2))) + ((sqrt (((- (Re h)) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2)) * <i> )) - t
proof end;

definition
let z1, z2, z3, z4, z be complex number ;
canceled;
redefine func Polynom z1,z2,z3,z4,z equals :: POLYEQ_3:def 5
(((z1 * (z ^3 )) + (z2 * (z ^2 ))) + (z3 * z)) + z4;
compatibility
for b1 being set holds
( b1 = Polynom z1,z2,z3,z4,z iff b1 = (((z1 * (z ^3 )) + (z2 * (z ^2 ))) + (z3 * z)) + z4 )
;
end;

:: deftheorem POLYEQ_3:def 4 :
canceled;

:: deftheorem defines Polynom POLYEQ_3:def 5 :
for z1, z2, z3, z4, z being complex number holds Polynom z1,z2,z3,z4,z = (((z1 * (z ^3 )) + (z2 * (z ^2 ))) + (z3 * z)) + z4;

theorem :: POLYEQ_3:38
canceled;

theorem :: POLYEQ_3:39
for z being Element of COMPLEX holds
( z |^ 3 = (z * z) * z & z |^ 3 = (z ^2 ) * z & z |^ 3 = z ^3 )
proof end;

theorem :: POLYEQ_3:40
for z1, z2, z being Element of COMPLEX st z1 <> 0 & Polynom z1,z2,0,0,z = 0 & not z = - (z2 / z1) holds
z = 0
proof end;

theorem :: POLYEQ_3:41
for z1, z3, z being Element of COMPLEX st z1 <> 0 & Polynom z1,0,z3,0,z = 0 holds
for s being Element of COMPLEX holds
( not s = - (z3 / z1) or z = 0 or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) )
proof end;

theorem :: POLYEQ_3:42
for z1, z2, z3, z being Element of COMPLEX st z1 <> 0 & Polynom z1,z2,z3,0,z = 0 holds
for s, h, t being Element of COMPLEX st s = - (z3 / z1) & h = ((z2 / (2 * z1)) ^2 ) - (z3 / z1) & t = z2 / (2 * z1) & not z = 0 & not z = ((sqrt (((Re h) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2)) + ((sqrt (((- (Re h)) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2)) * <i> )) - t & not z = ((- (sqrt (((Re h) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2))) + ((- (sqrt (((- (Re h)) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2))) * <i> )) - t & not z = ((sqrt (((Re h) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2)) + ((- (sqrt (((- (Re h)) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2))) * <i> )) - t holds
z = ((- (sqrt (((Re h) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2))) + ((sqrt (((- (Re h)) + (sqrt (((Re h) ^2 ) + ((Im h) ^2 )))) / 2)) * <i> )) - t
proof end;

theorem :: POLYEQ_3:43
for s, z1 being Element of COMPLEX holds (s - ((1 / 3) * z1)) ^2 = ((s ^2 ) + (((- (2 / 3)) * z1) * s)) + ((1 / 9) * (z1 ^2 )) ;

theorem :: POLYEQ_3:44
for z, s, z1 being Element of COMPLEX st z = s - ((1 / 3) * z1) holds
z ^3 = (((s ^3 ) - (z1 * (s ^2 ))) + (((1 / 3) * (z1 ^2 )) * s)) - ((1 / 27) * (z1 ^3 )) ;

theorem :: POLYEQ_3:45
canceled;

theorem Th46: :: POLYEQ_3:46
for z being Element of COMPLEX holds [*(|.z.| * (cos (Arg z))),(|.z.| * (sin (Arg z)))*] = [*|.z.|,0*] * [*(cos (Arg z)),(sin (Arg z))*]
proof end;

theorem :: POLYEQ_3:47
canceled;

theorem :: POLYEQ_3:48
canceled;

theorem :: POLYEQ_3:49
for z being complex number holds z |^ 2 = z * z by Lm2;

theorem Th50: :: POLYEQ_3:50
for n being Nat st n > 0 holds
0 |^ n = 0
proof end;

theorem Th51: :: POLYEQ_3:51
for x, y being Element of COMPLEX
for n being Element of NAT holds (x * y) |^ n = (x |^ n) * (y |^ n)
proof end;

theorem Th52: :: POLYEQ_3:52
for x being Real st x > 0 holds
for n being Element of NAT holds [*x,0*] |^ n = x to_power n
proof end;

theorem Th53: :: POLYEQ_3:53
for x being Real
for n being Element of NAT holds [*(cos x),(sin x)*] |^ n = (cos (n * x)) + ((sin (n * x)) * <i> )
proof end;

theorem :: POLYEQ_3:54
for z being Element of COMPLEX
for n being Element of NAT st ( z <> 0 or n > 0 ) holds
z |^ n = ((|.z.| to_power n) * (cos (n * (Arg z)))) + (((|.z.| to_power n) * (sin (n * (Arg z)))) * <i> )
proof end;

theorem Th55: :: POLYEQ_3:55
for n, k being Element of NAT
for x being Real st n <> 0 holds
[*(cos ((x + ((2 * PI ) * k)) / n)),(sin ((x + ((2 * PI ) * k)) / n))*] |^ n = (cos x) + ((sin x) * <i> )
proof end;

theorem Th56: :: POLYEQ_3:56
for z being complex number
for n, k being Element of NAT st n <> 0 holds
z = [*((n -root |.z.|) * (cos (((Arg z) + ((2 * PI ) * k)) / n))),((n -root |.z.|) * (sin (((Arg z) + ((2 * PI ) * k)) / n)))*] |^ n
proof end;

theorem :: POLYEQ_3:57
for z being Element of COMPLEX
for n being non empty Element of NAT
for k being Element of NAT holds [*((n -root |.z.|) * (cos (((Arg z) + ((2 * PI ) * k)) / n))),((n -root |.z.|) * (sin (((Arg z) + ((2 * PI ) * k)) / n)))*] is CRoot of n,z
proof end;

theorem :: POLYEQ_3:58
for z being complex number
for v being CRoot of 1,z holds v = z
proof end;

theorem :: POLYEQ_3:59
for n being non empty Nat
for v being CRoot of n,0 holds v = 0
proof end;

theorem :: POLYEQ_3:60
for n being non empty Element of NAT
for z being complex number
for v being CRoot of n,z st v = 0 holds
z = 0
proof end;

theorem :: POLYEQ_3:61
for n being non empty Element of NAT
for k being Element of NAT holds (cos (((2 * PI ) * k) / n)) + ((sin (((2 * PI ) * k) / n)) * <i> ) is CRoot of n,1
proof end;

theorem :: POLYEQ_3:62
canceled;

theorem :: POLYEQ_3:63
for z, s being Element of COMPLEX
for n being Element of NAT st s <> 0 & z <> 0 & n >= 1 & s |^ n = z |^ n holds
|.s.| = |.z.|
proof end;