
theorem
for F being non 2-characteristic Field,
    p being quadratic Polynomial of F
holds card(Roots p) = 2 iff DC p is non zero square
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;
L: 2 '*' a <> 0.F by ch2;
B: now assume B0: card(Roots p) = 2;
   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;
   B3: now assume (-b + w) * (2'*'a)" = (-b - w) * (2'*'a)";
       then Roots p = { (-b + w) * (2 '*' a)" }  by B2,ENUMSET1:29;
       hence contradiction by B0,CARD_2:42;
       end;
   now assume b^2 - 4 '*' a * c = 0.F;
       then w * w = 0.F by B1,O_RING_1:def 1;
       then w = 0.F by VECTSP_2:def 1;
       hence contradiction by B3;
       end;
   hence DC p is non zero square by A,defDC,B1;
   end;
now assume C0: DC p is non zero square;
   then b^2 - 4 '*' a * c is square by A,defDC;
   then consider w being Element of F such that
   C1: w^2 =  b^2 - 4 '*' a * c by O_RING_1:def 2;
   C2: Roots <%c,b,a%> = {(-b + w) * (2'*'a)",(-b - w) * (2'*'a)"} by C1,TC0;
   now assume (-b + w) * (2'*'a)" = (-b - w) * (2'*'a)";
     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 contradiction by C0,C1,A,defDC;
     end;
   hence card(Roots p) = 2 by A,C2,CARD_2:57;
   end;
hence thesis by B;
end;
