
theorem 2splitb:
Roots(F_Real,X^2-2) = {2-Root(2), -2-Root(2)}
proof
set F = F_Real, p = X^2-2;
set M = {2-Root(2), -2-Root(2)};
H: Roots(F,p) = {a where a is Element of F : a is_a_root_of p,F}
   by FIELD_4:def 4;
now let o be object;
   assume B: o in {2-Root(2), -2-Root(2)}; then
   reconsider c = o as Element of F_Real;
   per cases by B,TARSKI:def 2;
   suppose E0: o = 2-Root(2);
     Ext_eval(p,c)
        = 2-Root(2)|^(1+1) - 2 by E0,LKX2ext
       .= (2-Root(2)|^1 * 2-Root(2)|^1) - 2 by BINOM:10
       .= (2-Root(2)|^1 * 2-Root(2)) - 2 by BINOM:8
       .= (2-Root(2) * 2-Root(2)) - 2 by BINOM:8
       .= (sqrt 2)^2 - 2 by SQUARE_1:def 1
       .= 2 - 2 by SQUARE_1:def 2
       .= 0.F;
     then c is_a_root_of p,F by FIELD_4:def 2;
     hence o in Roots(F,p) by H;
     end;
   suppose E0: o = -2-Root(2);
     Ext_eval(p,c)
        = (-2-Root(2))|^(1+1) - 2 by E0,LKX2ext
       .= ((-2-Root(2))|^1 * (-2-Root(2))|^1) - 2 by BINOM:10
       .= ((-2-Root(2))|^1 * (-2-Root(2))) - 2 by BINOM:8
       .= ((-2-Root(2)) * (-2-Root(2))) - 2 by BINOM:8
       .= (2-Root(2) * 2-Root(2)) - 2
       .= (sqrt 2)^2 - 2 by SQUARE_1:def 1
       .= 2 - 2 by SQUARE_1:def 2
       .= 0.F;
     then c is_a_root_of p,F by FIELD_4:def 2;
     hence o in Roots(F,p) by H;
     end;
   end; then
B: M c= Roots(F_Real,X^2-2);
the carrier of Polynom-Ring F_Rat c= the carrier of Polynom-Ring F
  by FIELD_4:10; then
reconsider q = X^2-2 as Element of the carrier of Polynom-Ring F;
reconsider qq = q as non zero Element of the carrier of Polynom-Ring F_Rat;
H: 2= deg X^2-2 by FIELD_9:18 .= deg q by FIELD_4:20; then
reconsider q as non constant Element of the carrier of Polynom-Ring F_Real
   by RING_4:def 4;
C: card M = 2 by CARD_2:57,SQUARE_1:19;
D: card Roots(F_Real,X^2-2) = 2
   proof
   F: card(Roots q) <= 2 by H,RING_5:22;
   G: 2 <= card Roots(F_Real,qq) by C,B,NAT_1:43;
   Roots(F_Real,X^2-2) = Roots q by FIELD_7:13;
   hence thesis by F,G,XXREAL_0:1;
   end;
thus thesis by B,C,D,lemfinset;
end;
