
theorem TC0:
for F being non 2-characteristic Field
for a being non zero Element of F,
    b,c being Element of F
for w being Element of F st w^2 = b^2 - 4 '*' a * c
holds Roots <%c,b,a%> = { (-b + w) * (2 '*' a)", (-b - w) * (2 '*' a)" }
proof
let F be non 2-characteristic Field; let a be non zero Element of F;
let b,c be Element of F; let w be Element of F;
reconsider p = <%c,b,a%> as non zero Element of the carrier of Polynom-Ring F
   by POLYNOM3:def 10;
set r1 = (-b + w) * (2 '*' a)", r2 = (-b - w) * (2 '*' a)";
assume AS: w^2 = b^2 - 4 '*' a * c;
L: 2 '*' a <> 0.F by ch2;
C: now let o be object;
   assume C1: o in Roots p; then
   reconsider r = o as Element of F;
   r is_a_root_of p by C1,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;
   then per cases by AS,REALALG2:10;
   suppose w = 2'*'a*r + b; then
     -b + w = 2'*'a*r + (b + -b) by RLVECT_1:def 3
           .= 2'*'a*r + 0.F by RLVECT_1:5
           .= (2'*'a)*r; then
     (2'*'a)" * (-b + w) = ((2'*'a)" * (2'*'a)) * r by GROUP_1:def 3
                        .= 1.F * r by L,VECTSP_1:def 10; then
     r = (-b + w) * (2 '*' a)" by GROUP_1:def 12;
     hence o in { r1, r2 } by TARSKI:def 2;
     end;
   suppose w = -(2'*'a*r + b); then
     -b + -w = 2'*'a*r + (b + -b) by RLVECT_1:def 3
            .= 2'*'a*r + 0.F by RLVECT_1:5
            .= (2'*'a)*r;then
     (2'*'a)" * (-b + -w) = ((2'*'a)" * (2'*'a)) * r by GROUP_1:def 3
                         .= 1.F * r by L,VECTSP_1:def 10; then
     r = (-b - w) * (2 '*' a)" by GROUP_1:def 12;
     hence o in { r1, r2 } by TARSKI:def 2;
     end;
   end;
now let o be object;
  assume o in { r1, r2 }; then
  per cases by TARSKI:def 2;
  suppose D: o = r1;
    r1 is_a_root_of p by AS,lemeval;
    hence o in Roots p by D,POLYNOM5:def 10;
    end;
  suppose D: o = r2;
    r2 is_a_root_of p by AS,lemeval;
    hence o in Roots p by D,POLYNOM5:def 10;
    end;
 end;
hence thesis by C,TARSKI:2;
end;
