
theorem 2splita:
X^2-2 = X-2-Root(2) * X+2-Root(2)
proof
set F = F_Real, a = 2-Root(2), b = -2-Root(2);
A: X-a = rpoly(1,a) by FIELD_9:def 2 .= <%-a, 1.F%> by RING_5:10;
B: X-b = rpoly(1,b) by FIELD_9: def 2 .= <%-b, 1.F%> by RING_5:10;
C: X-b = rpoly(1,-2-Root(2)) by FIELD_9:def 2 .= X+2-Root(2) by FIELD_9:def 3;
D: (X-a) * (X-b) = <%-a, 1.F%> *' <%-b, 1.F%> by A,B,POLYNOM3:def 10
     .= <%(-a)*(-b),(1.F)*(-b)+(1.F)*(-a),(1.F)*(1.F)%> by FIELD_9:24
     .= <%a*b,-b+-a,1.F%>;
H: a * b = -((sqrt 2) * (sqrt 2))
        .= -((sqrt 2)^2) by SQUARE_1:def 1
        .= -(1.F_Rat + 1.F_Rat) by SQUARE_1:def 2,GAUSSINT:def 14;
the carrier of Polynom-Ring F_Rat c= the carrier of Polynom-Ring F
  by FIELD_4:10; then
X^2-2 in the carrier of Polynom-Ring F; then
reconsider p = X^2-2 as Polynomial of F by POLYNOM3:def 10;
E: dom p = NAT by FUNCT_2:def 1
        .= dom <%a*b,-b+-a,1.F%> by FUNCT_2:def 1;
now let o be object;
  assume o in dom <%a*b,-b+-a,1.F%>;
  then reconsider i = o as Element of NAT;
  i <= 2 implies i = 0 or ... or i = 2; then
  per cases;
  suppose C: i = 0; then
    <%a*b,-b+-a,1.F%>.i = a * b by FIELD_9:16
                       .= p.i by C,H,FIELD_9:16;
    hence <%a*b,-b+-a,1.F%>.o = p.o;
    end;
  suppose C: i = 1; then
    <%a*b,-b+-a,1.F%>.i = -b + -a by FIELD_9:16
                       .= p.i by C,FIELD_9:16,GAUSSINT:def 14;
    hence <%a*b,-b+-a,1.F%>.o = p.o;
    end;
  suppose C: i = 2; then
    <%a*b,-b+-a,1.F%>.i = 1.F by FIELD_9:16
                       .= p.i by C,FIELD_9:16,GAUSSINT:def 14;
    hence <%a*b,-b+-a,1.F%>.o = p.o;
    end;
  suppose i > 2; then
    C1: i >= 2 + 1 by NAT_1:13; then
    <%a*b,-b+-a,1.F%>.i = 0.F by FIELD_9:16
                       .= p.i by C1,FIELD_9:16,GAUSSINT:def 14;
    hence <%a*b,-b+-a,1.F%>.o = p.o;
    end;
  end;
then p = <%a*b,-b+-a,1.F%> by E
      .= (X-2-Root(2)) * (X+2-Root(2)) by C,D;
hence thesis;
end;
