
theorem evalext2:
for z being Element of F_Complex holds Ext_eval(X^3-2,z) = z|^3 - 2
proof
let x be Element of F_Complex;
set R = F_Complex, p = X^3-2, t = -(1.R + 1.R);
A0: t = -(1 + 1) by COMPLFLD:2,COMPLFLD:8,COMPLEX1:def 4;
consider F being FinSequence of the carrier of F_Complex such that
A1: Ext_eval(X^3-2,x) = Sum F and
A2: len F = len p and
A3: for n be Element of NAT st n in dom F holds
    F.n = In(p.(n-'1),R) * (power R).(x,n-'1) by ALGNUM_1:def 1;
A5: F.1 = In(p.(1-'1),R) * (power R).(x,1-'1) by A3,A2,LL1,FINSEQ_3:25
       .= In(p.0,R) * (power R).(x,1-'1) by XREAL_1:232
       .= p.0 * (power R).(x,0) by XREAL_1:232
       .= p.0 * 1_R by GROUP_1:def 7
       .= t by COMPLFLD:8,COMPLEX1:def 4,COMPLFLD:2,LL0;
A6: 2-'1 = 2-1 by XREAL_0:def 2;
A7: F.2 = In(p.(2-'1),R) * (power R).(x,2-'1) by A3,A2,LL1,FINSEQ_3:25
       .= 0.R * (power R).(x,1) by COMPLFLD:7,A6,LL0;
A8: 3-'1 = 3-1 by XREAL_0:def 2;
A9: F.3 = In(p.(3-'1),R) * (power R).(x,3-'1) by A3,A2,LL1,FINSEQ_3:25
       .= 0.R * (power R).(x,2) by COMPLFLD:7,A8,LL0;
A10: 4-'1 = 4-1 by XREAL_0:def 2;
A11: F.4 = In(p.(4-'1),R) * (power R).(x,4-'1) by A3,A2,LL1,FINSEQ_3:25
        .= x|^3 by A10,LL0,BINOM:def 2;
F = <*t,0.R,0.R,x|^3*> by A2,LL1,A5,A7,A9,A11,FINSEQ_4:76
 .= <*t,0.R,0.R*> ^ <*x|^3*> by FINSEQ_4:74;
hence Ext_eval(p,x)
   = Sum <*t,0.R,0.R*> + Sum <*x|^3*> by A1,RLVECT_1:41
  .= Sum <*t,0.R,0.R*> + x|^3 by RLVECT_1:44
  .= x |^ 3 - 2 by A0,RLVECT_1:72;
end;
