
theorem z23:
X^2+X = X_ *' (X-1) & Roots X^2+X = { 0.(Z/2), 1.(Z/2) }
proof
A: X_ *' X-1 = rpoly(1,-0.(Z/2)) *' <%1.(Z/2),--1.(Z/2)%> by RING_5:10
           .= rpoly(1,0.(Z/2)) *' rpoly(1,1.(Z/2)) by cz2a,RING_5:10;
hence B: X_ *' X-1
            = <%0.(Z/2)*1.(Z/2),-(0.(Z/2)+1.(Z/2)),1.(Z/2)%> by lemred3z
           .= X^2+X by FIELD_3:4,RLVECT_1:def 10;
Roots rpoly(1,1.(Z/2)) = { 1.(Z/2) } &
Roots rpoly(1,0.(Z/2)) = { 0.(Z/2) } by RING_5:18;
then Roots(rpoly(1,0.(Z/2)) *' rpoly(1,1.(Z/2)))
   = { 0.(Z/2) } \/ { 1.(Z/2) } by UPROOTS:23
  .= { 0.(Z/2), 1.(Z/2) } by ENUMSET1:1;
hence thesis by A,B;
end;
