reserve
F for non 2-characteristic non quadratic_complete polynomial_disjoint Field;
reserve
p for non DC-square quadratic Element of the carrier of Polynom-Ring F;

theorem Z5:
p = @(LC p,FAdj(F,{sqrt(DC p)})) * (X-(Root1 p)) *' (X-(Root2 p))
proof
set E = FAdj(F,{sqrt(DC p)});
K: F is Subring of E by FIELD_4:def 1; then
H: E is non 2-characteristic;
consider a being non zero Element of F, b,c being Element of F such that
A: p = <%c,b,a%> by qua5;
I: @(c,E) = c & @(b,E) = b & @(a,E) = a by FIELD_7:def 4; then
B: p = <%@(c,E),@(b,E),@(a,E)%> by A,eval2;
E: now assume E1: @(a,E) is zero;
   a = @(a,E) by FIELD_7:def 4 .= 0.F by E1,K,C0SP1:def 3;
   hence contradiction;
   end;
C: Root1 p = (-@(b,E) + (RootDC p)) * (2 '*' @(a,E))" by E,B,Z2;
D: Root2 p = (-@(b,E) - (RootDC p)) * (2 '*' @(a,E))" by E,B,Z2;
J: (RootDC p)^2 = (RootDC p) * (RootDC p) by O_RING_1:def 1
               .= DC p by Z1
               .= b^2 - 4 '*' a * c by A,defDC;
   J1: b^2 = b * b by O_RING_1:def 1
          .= @(b,E) * @(b,E) by I,K,FIELD_6:16
          .= @(b,E)^2 by O_RING_1:def 1;
       4 '*' a * c = 4 '*' (a * c) by REALALG2:5
                  .= 4 '*' (@(a,E) * @(c,E)) by Z3,I,K,FIELD_6:16
                  .= 4 '*' @(a,E) * @(c,E) by REALALG2:5; then
       - (4 '*' a * c) = - (4 '*' @(a,E) * @(c,E)) by K,FIELD_6:17;
   then
F: b^2 - 4 '*' a * c = @(b,E)^2 - 4 '*' @(a,E) * @(c,E) by K,J1,FIELD_6:15;
   len p = 3 by A,qua3; then
G: len p -' 1 = 3 - 1 by XREAL_0:def 2;
   @(a,E) = a by FIELD_7:def 4
         .= p.2 by A,qua1
         .= LC p by G,RATFUNC1:def 6
         .= @(LC p,E) by FIELD_7:def 4;
hence thesis by F,J,H,E,D,C,B,lemred;
end;
