
theorem z22:
X^2+1 = (X-1) *' (X-1) & Roots X^2+1 = { 1.(Z/2) }
proof
A: X-1 *' X-1 = rpoly(1,-1.(Z/2)) *' <%1.(Z/2),--1.(Z/2)%> by RING_5:10
             .= rpoly(1,1.(Z/2)) *' rpoly(1,1.(Z/2)) by cz2a,RING_5:10;
hence B: X-1 *' X-1
       = <%1.(Z/2)*1.(Z/2),-(1.(Z/2)+1.(Z/2)),1.(Z/2)%> by lemred3z
      .= X^2+1 by FIELD_3:4;
Roots rpoly(1,1.(Z/2)) = { 1.(Z/2) } by RING_5:18;
then Roots(rpoly(1,1.(Z/2)) *' rpoly(1,1.(Z/2)))
   = { 1.(Z/2) } \/ { 1.(Z/2) } by UPROOTS:23
  .= { 1.(Z/2) };
hence thesis by A,B;
end;
