theorem
  A* = {x} implies x = <%>E
proof
  assume that
A1: A* = {x} and
A2: x <> <%>E;
  reconsider a = x as Element of E^omega by A1,ZFMISC_1:31;
  x in A* by A1,ZFMISC_1:31;
  then
A3: a ^ a in A* by Th45;
  a ^ a <> x by A2,Th11;
  hence thesis by A1,A3,TARSKI:def 1;
end;
