reserve
F for non 2-characteristic non quadratic_complete polynomial_disjoint Field;

theorem Fi1a:
for a being non square Element of F holds (X-(sqrt a)) *' (X+(sqrt a)) = X^2-a
proof
let a be non square Element of F;
set E = embField(canHomP X^2-a);
H: F is Subring of E by FIELD_4:def 1; then
   the carrier of F c= the carrier of E by C0SP1:def 3; then
reconsider b = a as Element of E;
A: (sqrt a) * (-(sqrt a)) = -((sqrt a) * (sqrt a)) by VECTSP_1:8 .= -b by m1;
   -((sqrt a) + -(sqrt a))  = -0.E by RLVECT_1: 5 .= 0.E; then
B: (X-(sqrt a)) *' (X--(sqrt a)) = <%-b,0.E,1.E%> by A,lemred3z;
   0.E = 0.F & 1.E = 1.F by H,C0SP1:def 3;
hence thesis by B,H,FIELD_6:17;
end;
