:: POLYEQ_1 semantic presentation
:: deftheorem Def1 defines Polynom POLYEQ_1:def 1 :
theorem Th1: :: POLYEQ_1:1
theorem Th2: :: POLYEQ_1:2
theorem Th3: :: POLYEQ_1:3
:: deftheorem Def2 defines Polynom POLYEQ_1:def 2 :
theorem Th4: :: POLYEQ_1:4
for b
1, b
2, b
3, b
4, b
5, b
6 being
complex number holds
( ( for b
7 being
real number holds
Polynom b
1,b
2,b
3,b
7 = Polynom b
4,b
5,b
6,b
7 ) implies ( b
1 = b
4 & b
2 = b
5 & b
3 = b
6 ) )
theorem Th5: :: POLYEQ_1:5
for b
1, b
2, b
3 being
real number holds
( b
1 <> 0 &
delta b
1,b
2,b
3 >= 0 implies for b
4 being
real number holds
not (
Polynom b
1,b
2,b
3,b
4 = 0 & not b
4 = ((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1) & not b
4 = ((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1) ) )
theorem Th6: :: POLYEQ_1:6
theorem Th7: :: POLYEQ_1:7
theorem Th8: :: POLYEQ_1:8
theorem Th9: :: POLYEQ_1:9
theorem Th10: :: POLYEQ_1:10
:: deftheorem Def3 defines Quard POLYEQ_1:def 3 :
theorem Th11: :: POLYEQ_1:11
:: deftheorem Def4 defines Polynom POLYEQ_1:def 4 :
registration
let c
1, c
2, c
3, c
4, c
5 be
complex number ;
cluster Polynom a
1,a
2,a
3,a
4,a
5 -> complex ;
coherence
Polynom c1,c2,c3,c4,c5 is complex
;
end;
registration
let c
1, c
2, c
3, c
4, c
5 be
real number ;
cluster Polynom a
1,a
2,a
3,a
4,a
5 -> complex real ;
coherence
Polynom c1,c2,c3,c4,c5 is real
;
end;
definition
let c
1, c
2, c
3, c
4, c
5 be
Real;
redefine func Polynom as
Polynom c
1,c
2,c
3,c
4,c
5 -> Real;
coherence
Polynom c1,c2,c3,c4,c5 is Real
by XREAL_0:def 1;
end;
theorem Th12: :: POLYEQ_1:12
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
real number holds
( ( for b
9 being
real number holds
Polynom b
1,b
2,b
3,b
4,b
9 = Polynom b
5,b
6,b
7,b
8,b
9 ) implies ( b
1 = b
5 & b
2 = b
6 & b
3 = b
7 & b
4 = b
8 ) )
:: deftheorem Def5 defines Tri POLYEQ_1:def 5 :
for b
1, b
2, b
3, b
4, b
5 being
real number holds
Tri b
1,b
3,b
4,b
5,b
2 = b
1 * (((b2 - b3) * (b2 - b4)) * (b2 - b5));
registration
let c
1, c
2, c
3, c
4, c
5 be
real number ;
cluster Tri a
1,a
3,a
4,a
5,a
2 -> real ;
coherence
Tri c1,c3,c4,c5,c2 is real
;
end;
definition
let c
1, c
2, c
3, c
4, c
5 be
Real;
redefine func Tri as
Tri c
1,c
3,c
4,c
5,c
2 -> Real;
coherence
Tri c1,c3,c4,c5,c2 is Real
by XREAL_0:def 1;
end;
theorem Th13: :: POLYEQ_1:13
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
real number holds
( b
1 <> 0 & ( for b
8 being
real number holds
Polynom b
1,b
2,b
3,b
4,b
8 = Tri b
1,b
5,b
6,b
7,b
8 ) implies ( b
2 / b
1 = - ((b5 + b6) + b7) & b
3 / b
1 = ((b5 * b6) + (b6 * b7)) + (b5 * b7) & b
4 / b
1 = - ((b5 * b6) * b7) ) )
theorem Th14: :: POLYEQ_1:14
theorem Th15: :: POLYEQ_1:15
for b
1, b
2, b
3, b
4, b
5 being
real number holds
( b
1 <> 0 &
Polynom b
1,b
2,b
3,b
4,b
5 = 0 implies for b
6, b
7, b
8, b
9, b
10 being
real number holds
( b
10 = b
5 + (b2 / (3 * b1)) & b
9 = - (b2 / (3 * b1)) & b
6 = b
2 / b
1 & b
7 = b
3 / b
1 & b
8 = b
4 / b
1 implies
((b10 |^ 3) + ((((3 * b9) + b6) * (b10 ^2 )) + ((((3 * (b9 ^2 )) + (2 * (b6 * b9))) + b7) * b10))) + (((b9 |^ 3) + (b6 * (b9 ^2 ))) + ((b7 * b9) + b8)) = 0 ) )
theorem Th16: :: POLYEQ_1:16
for b
1, b
2, b
3, b
4, b
5 being
real number holds
( b
1 <> 0 &
Polynom b
1,b
2,b
3,b
4,b
5 = 0 implies for b
6, b
7, b
8, b
9, b
10 being
real number holds
( b
10 = b
5 + (b2 / (3 * b1)) & b
9 = - (b2 / (3 * b1)) & b
6 = b
2 / b
1 & b
7 = b
3 / b
1 & b
8 = b
4 / b
1 implies
(((b10 |^ 3) + (0 * (b10 ^2 ))) + (((((3 * b1) * b3) - (b2 ^2 )) / (3 * (b1 ^2 ))) * b10)) + ((2 * ((b2 / (3 * b1)) |^ 3)) + ((((3 * b1) * b4) - (b2 * b3)) / (3 * (b1 ^2 )))) = 0 ) )
theorem Th17: :: POLYEQ_1:17
for b
1, b
2, b
3, b
4, b
5 being
real number holds
(
(((b1 |^ 3) + (0 * (b1 ^2 ))) + (((((3 * b2) * b3) - (b4 ^2 )) / (3 * (b2 ^2 ))) * b1)) + ((2 * ((b4 / (3 * b2)) |^ 3)) + ((((3 * b2) * b5) - (b4 * b3)) / (3 * (b2 ^2 )))) = 0 implies for b
6, b
7 being
real number holds
( b
6 = (((3 * b2) * b3) - (b4 ^2 )) / (3 * (b2 ^2 )) & b
7 = (2 * ((b4 / (3 * b2)) |^ 3)) + ((((3 * b2) * b5) - (b4 * b3)) / (3 * (b2 ^2 ))) implies
Polynom 1,0,b
6,b
7,b
1 = 0 ) ) ;
theorem Th18: :: POLYEQ_1:18
theorem Th19: :: POLYEQ_1:19
theorem Th20: :: POLYEQ_1:20
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 <> 0 &
delta b
1,b
2,b
3 > 0 &
Polynom 0,b
1,b
2,b
3,b
4 = 0 & not b
4 = ((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1) & not b
4 = ((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1) )
theorem Th21: :: POLYEQ_1:21
for b
1, b
2, b
3, b
4, b
5, b
6 being
real number holds
( b
1 <> 0 & b
2 = b
3 / b
1 & b
4 = b
5 / b
1 &
Polynom b
1,0,b
3,b
5,b
6 = 0 implies for b
7, b
8 being
real number holds
not ( b
6 = b
7 + b
8 &
((3 * b8) * b7) + b
2 = 0 & not b
6 = (3 -root ((- (b5 / (2 * b1))) + (sqrt (((b5 ^2 ) / (4 * (b1 ^2 ))) + ((b3 / (3 * b1)) |^ 3))))) + (3 -root ((- (b5 / (2 * b1))) - (sqrt (((b5 ^2 ) / (4 * (b1 ^2 ))) + ((b3 / (3 * b1)) |^ 3))))) & not b
6 = (3 -root ((- (b5 / (2 * b1))) + (sqrt (((b5 ^2 ) / (4 * (b1 ^2 ))) + ((b3 / (3 * b1)) |^ 3))))) + (3 -root ((- (b5 / (2 * b1))) + (sqrt (((b5 ^2 ) / (4 * (b1 ^2 ))) + ((b3 / (3 * b1)) |^ 3))))) & not b
6 = (3 -root ((- (b5 / (2 * b1))) - (sqrt (((b5 ^2 ) / (4 * (b1 ^2 ))) + ((b3 / (3 * b1)) |^ 3))))) + (3 -root ((- (b5 / (2 * b1))) - (sqrt (((b5 ^2 ) / (4 * (b1 ^2 ))) + ((b3 / (3 * b1)) |^ 3))))) ) )
theorem Th22: :: POLYEQ_1:22
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 <> 0 &
delta b
1,b
2,b
3 >= 0 &
Polynom b
1,b
2,b
3,0,b
4 = 0 & not b
4 = 0 & not b
4 = ((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1) & not b
4 = ((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1) )
theorem Th23: :: POLYEQ_1:23