
theorem z21:
X^2 = X_ *' X_ & Roots X^2 = { 0.(Z/2) }
proof
A: X_ *' X_ = rpoly(1,-0.(Z/2)) *' <%0.(Z/2),1.(Z/2)%> by RING_5:10
         .= rpoly(1,0.(Z/2)) *' rpoly(1,-0.(Z/2)) by RING_5:10;
thus B: X_ *' X_ = <%0.(Z/2)*0.(Z/2),-(0.(Z/2)+0.(Z/2)),1.(Z/2)%> by lemred3
              .= X^2;
Roots rpoly(1,0.(Z/2)) = { 0.(Z/2) } by RING_5:18;
then Roots(rpoly(1,0.(Z/2)) *' rpoly(1,0.(Z/2)))
   = { 0.(Z/2) } \/ { 0.(Z/2) } by UPROOTS:23
  .= { 0.(Z/2) };
hence thesis by A,B;
end;
