
theorem TC1:
for F being non 2-characteristic Field,
    p being quadratic Polynomial of F holds Roots p <> {} iff DC p is 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;
B: now assume DC p is square; then
   b^2 - 4 '*' a * c is being_a_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;
   (-b + w) * (2 '*' a)" is_a_root_of p by A,B1,lemeval;
   hence Roots p <> {} by POLYNOM5:def 10;
   end;
now assume Roots p <> {};
   then b^2 - 4 '*' a * c is square by A,lemeval2;
   hence DC p is square by A,defDC;
   end;
hence thesis by B;
end;
