
theorem
for F being non 2-characteristic Field,
    p being quadratic Polynomial of F holds card(Roots p) = 1 iff DC p = 0.F
proof
let F be non 2-characteristic Field, p be quadratic Polynomial of F;
consider a being non zero Element of F, b,c being Element of F such that
A: p = <%c,b,a%> by qua5;
set r = the Element of Roots p;
B: now assume DC p = 0.F; then
   B0: b^2 - 4 '*' a * c = 0.F by A,defDC; then
   consider w being Element of F such that
   B1: b^2 - 4 '*' a * c = w^2 by O_RING_1:def 2;
   B2: Roots p = { (-b + w) * (2 '*' a)", (-b - w) * (2 '*' a)" } by B1,A,TC0;
   0.F = w * w by B0,B1,O_RING_1:def 1; then
   w = 0.F by VECTSP_2:def 1; then
   Roots p = { (-b + w) * (2 '*' a)" } by B2,ENUMSET1:29;
   hence card(Roots p) = 1 by CARD_1:30;
   end;
L: 2 '*' a <> 0.F by ch2;
now assume B0: card(Roots p) = 1;
   then Roots p <> {};
   then DC p is square by TC1;
   then b^2 - 4 '*' a * c is square by A,defDC; then
   consider w being Element of F such that
   B1: b^2 - 4 '*' a * c = w^2 by O_RING_1:def 2;
   B2: Roots p = { (-b + w) * (2 '*' a)", (-b - w) * (2 '*' a)" } by B1,A,TC0;
   (-b + w) * (2 '*' a)" = (-b - w) * (2 '*' a)" by B0,B2,CARD_2:57;
   then ((-b + w) * (2 '*' a)") * (2 '*' a)
      = (-b - w) * ((2 '*' a)" * (2 '*' a)) by GROUP_1:def 3
     .= (-b - w) * 1.F by L,VECTSP_1:def 10;
   then - b - w = (-b + w) * ((2 '*' a)" * (2 '*' a) ) by GROUP_1:def 3
               .= (-b + w) * 1.F by L,VECTSP_1:def 10;
   then b + ((-b) - w) = (b + (-b)) + w by RLVECT_1:def 3
                      .= 0.F + w by RLVECT_1:5;
   then w = (b + (-b)) + -w by RLVECT_1:def 3 .= 0.F + -w by RLVECT_1:5;
   then w + w = 0.F by RLVECT_1:5;
   then 2 '*' w = 0.F by RING_5:2;
   then w = 0.F by ch2;
   hence DC p = 0.F by B1,A,defDC;
   end;
hence thesis by B;
end;
