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

theorem m1:
for a being non square Element of F holds (sqrt a) * (sqrt a) = a
proof
let a be non square Element of F;
B: F is Subring of embField(canHomP X^2-a) by FIELD_4:def 1; then
the carrier of F c= the carrier of embField(canHomP X^2-a) by C0SP1:def 3; then
reconsider b = a as Element of embField(canHomP X^2-a);
H: -a = -b &
   0.F = 0.embField(canHomP X^2-a) &
   1.F = 1.embField(canHomP X^2-a) by FIELD_6:17,B,C0SP1:def 3;
0.embField(canHomP X^2-a) = 0.F by B,C0SP1:def 3
   .= Ext_eval(X^2-a,sqrt a) by FIELD_5:17
   .= eval(X^2-b,sqrt a) by H,FIELD_4:26
   .= (-b) + (0.embField(canHomP X^2-a)) * (sqrt a)
           + (1.embField(canHomP X^2-a)) * (sqrt a)^2 by evalq
   .= (sqrt a) * (sqrt a) - b by O_RING_1:def 1;
hence thesis by RLVECT_1:21;
end;
