
theorem lemeval2:
for F being Field
for a being non zero Element of F, b,c being Element of F
st Roots <%c,b,a%> <> {} holds b^2 - 4 '*' a * c is square
proof
let F be Field;
let a be non zero Element of F, b,c be Element of F;
set p = <%c,b,a%>, r = the Element of Roots <%c,b,a%>;
assume AS: Roots <%c,b,a%> <> {}; then
r in Roots p; then
reconsider r as Element of F;
r is_a_root_of p by AS,POLYNOM5:def 10; then
0.F = c + b * r + a * r^2 by evalq
   .= a * r^2 + b * r  + c by RLVECT_1:def 3
   .= a * r^2 + r * b  + c by GROUP_1:def 12; then
0.F = (4 '*' a) * (a * r^2 + r*b + c)
   .= (4 '*' a) * (a * r^2 + r*b) + (4'*'a)*c by VECTSP_1:def 2
   .= ((4 '*' a) * (a * r^2) + (4'*'a)*(r*b)) + (4'*'a)*c by VECTSP_1:def 2
   .= (4 '*' a) * (a * r^2) + (4'*'a)*r*b + (4'*'a)*c by GROUP_1:def 3
   .= ((4 '*' a) * a) * r^2 + 4'*'a*r*b + 4'*'a*c by GROUP_1:def 3
   .= (4 '*' (a * a)) * r^2 + 4'*'a*r*b + 4'*'a*c by REALALG2:5
   .= ((2*2) '*' a^2) * r^2 + 4'*'a*r*b + 4'*'a*c by O_RING_1:def 1
   .= ((2^2) '*' a^2) * r^2 + 4'*'a*r*b + 4'*'a*c by SQUARE_1:def 1
   .= (2'*'a)^2 * r^2 + 4'*'a*r*b + 4'*'a*c by ch1
   .= (2'*'a*r)^2 + 4'*'a*r*b + 4'*'a*c by ch0; then
- 4'*'a*c = (2'*'a*r)^2 + 4'*'a*r*b + 4'*'a*c - 4'*'a*c
         .= (2'*'a*r)^2 + 4'*'a*r*b + (4'*'a*c - 4'*'a*c) by RLVECT_1:def 3
         .= (2'*'a*r)^2 + 4'*'a*r*b + 0.F by RLVECT_1:15; then
b^2 + -4'*'a*c = (2'*'a*r)^2 + ((2*2)'*'(a*r))*b + b^2 by REALALG2:5
              .= (2'*'a*r)^2 + (2'*'(2'*'(a*r)))*b + b^2 by RING_3:65
              .= (2'*'a*r)^2 + (2'*'(2'*'a*r))*b + b^2 by REALALG2:5
              .= (2'*'a*r + b)^2 by REALALG2:7;
hence thesis;
end;
