reserve
F for non 2-characteristic non quadratic_complete polynomial_disjoint Field;
reserve
p for non DC-square quadratic Element of the carrier of Polynom-Ring F;

theorem
Root1 p <> Root2 p
proof
set E = FAdj(F,{sqrt(DC p)}), w = RootDC p;
consider a1 being non zero Element of F,
         b1,c1 being Element of F such that
A: p = <%c1,b1,a1%> by qua5;
set a = @(a1,E), b = @(b1,E), c = @(c1,E);
I: a = a1 & b = b1 & c = c1 by FIELD_7:def 4; then
B: p = <%c,b,a%> by A,eval2;
J: E is Subring of embField(canHomP X^2-(DC p)) by FIELD_4:def 1;
K: F is Subring of E by FIELD_4:def 1;
E: a is non zero by I,K,C0SP1:def 3;
M: E is non 2-characteristic by K; then
L: 2 '*' a <> 0.E by E,ch2;
now assume Root1 p = Root2 p; then
     (-b + w)*(2'*'a)" = Root2 p by E,B,Z2 .= (-b - w)*(2'*'a)" by E,B,Z2;
     then ((-b + w) * (2 '*' a)") * (2 '*' a)
        = (-b - w) * ((2 '*' a)" * (2 '*' a)) by GROUP_1:def 3
       .= (-b - w) * 1.E by L,VECTSP_1:def 10;
     then - b - w = (-b + w) * ((2 '*' a)" * (2 '*' a) ) by GROUP_1:def 3
                 .= (-b + w) * 1.E by L,VECTSP_1:def 10;
     then b + ((-b) - w) = (b + (-b)) + w by RLVECT_1:def 3
                        .= 0.E + w by RLVECT_1:5;
     then w = (b + (-b)) + -w by RLVECT_1:def 3 .= 0.E + -w by RLVECT_1:5;
     then w + w = 0.E by RLVECT_1:5;
     then 2 '*' w = 0.E by RING_5:2;
     then w = 0.E by M,ch2
           .= 0.embField(canHomP X^2-(DC p)) by J,C0SP1:def 3;
     hence contradiction;
     end;
hence thesis;
end;
