
theorem pz2:
the set of all p where p is quadratic Polynomial of Z/2
  = { X^2, X^2+1, X^2+X, X^2+X+1 }
proof
set M = { <%0.(Z/2), 0.(Z/2), 1.(Z/2)%>, <%1.(Z/2), 0.(Z/2), 1.(Z/2)%>,
          <%0.(Z/2), 1.(Z/2), 1.(Z/2)%>, <%1.(Z/2), 1.(Z/2), 1.(Z/2)%> };
A: now let o be object;
   assume o in M;
   then o is quadratic Polynomial of Z/2 by ENUMSET1:def 2;
   hence o in the set of all p where p is quadratic Polynomial of Z/2;
   end;
now let o be object;
  assume o in the set of all q where q is quadratic Polynomial of Z/2;
  then consider p being quadratic Polynomial of Z/2 such that A1: o = p;
  consider b,c being Element of Z/2 such that
  A2: p = <%c,b,1.(Z/2)%> by qua5a;
  per cases by cz2,TARSKI:def 2;
  suppose A3: b = 1.(Z/2);
    per cases by cz2,TARSKI:def 2;
    suppose c = 1.(Z/2);
      hence o in M by A3,A2,A1,ENUMSET1:def 2;
      end;
    suppose c = 0.(Z/2);
      hence o in M by A3,A2,A1,ENUMSET1:def 2;
      end;
    end;
  suppose A3: b = 0.(Z/2);
    per cases by cz2,TARSKI:def 2;
    suppose c = 1.(Z/2);
      hence o in M by A3,A2,A1,ENUMSET1:def 2;
      end;
    suppose c = 0.(Z/2);
      hence o in M by A3,A2,A1,ENUMSET1:def 2;
      end;
    end;
  end;
hence thesis by A,TARSKI:2;
end;
