let x be Element of F_Real; Ext_eval (X^2-2,x) = (x |^ 2) - 2
set R = F_Real ;
set p = X^2-2 ;
set t = - ((1. F_Real) + (1. F_Real));
F_Rat is Subring of F_Real
by FIELD_4:def 1;
then J:
1. F_Rat = 1. F_Real
by C0SP1:def 3;
consider F being FinSequence of the carrier of F_Real such that
A1:
Ext_eval (X^2-2,x) = Sum F
and
A2:
len F = len X^2-2
and
A3:
for n being Element of NAT st n in dom F holds
F . n = (In ((X^2-2 . (n -' 1)),F_Real)) * ((power F_Real) . (x,(n -' 1)))
by ALGNUM_1:def 1;
B1:
X^2-2 . 0 = - ((1. F_Real) + (1. F_Real))
by J, FIELD_9:16;
B2:
X^2-2 . 1 = 0. F_Real
by GAUSSINT:def 14, FIELD_9:16;
B3:
X^2-2 . 2 = 1. F_Real
by GAUSSINT:def 14, FIELD_9:16;
BB:
deg X^2-2 = 2
by FIELD_9:18;
B0:
deg X^2-2 = (len X^2-2) - 1
by HURWITZ:def 2;
then B4:
len F = 3
by A2, BB;
then A5: F . 1 =
(In ((X^2-2 . (1 -' 1)),F_Real)) * ((power F_Real) . (x,(1 -' 1)))
by A3, FINSEQ_3:25
.=
(In ((X^2-2 . 0),F_Real)) * ((power F_Real) . (x,(1 -' 1)))
by XREAL_1:232
.=
(X^2-2 . 0) * ((power F_Real) . (x,0))
by XREAL_1:232
.=
(- ((1. F_Real) + (1. F_Real))) * (1_ F_Real)
by B1, GROUP_1:def 7
;
A6:
2 -' 1 = 2 - 1
by XREAL_0:def 2;
A7: F . 2 =
(In ((X^2-2 . (2 -' 1)),F_Real)) * ((power F_Real) . (x,(2 -' 1)))
by B4, A3, FINSEQ_3:25
.=
(0. F_Real) * (x |^ 1)
by B2, A6
;
A8:
3 -' 1 = 3 - 1
by XREAL_0:def 2;
A9: F . 3 =
(In ((X^2-2 . (3 -' 1)),F_Real)) * ((power F_Real) . (x,(3 -' 1)))
by B0, A3, A2, BB, FINSEQ_3:25
.=
x |^ 2
by B3, A8, BINOM:def 2
;
F = <*(- ((1. F_Real) + (1. F_Real))),(0. F_Real),(x |^ 2)*>
by B0, A2, BB, A5, A7, A9, FINSEQ_1:45;
hence Ext_eval (X^2-2,x) =
((- ((1. F_Real) + (1. F_Real))) + (0. F_Real)) + (x |^ 2)
by A1, RLVECT_1:46
.=
(x |^ 2) - 2
;
verum