reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, a1, a2, b for Element of E^omega;
reserve i, k, l, m, n for Nat;

theorem Th38:
  A |^.. k = {x} implies x = <%>E
proof
  assume that
A1: A |^.. k = {x} and
A2: x <> <%>E;
  reconsider a = x as Element of E^omega by A1,ZFMISC_1:31;
  x in A |^.. k by A1,ZFMISC_1:31;
  then
A3: a ^ a in A |^.. (k + k) by Th37;
A4: A |^.. (k + k) c= A |^.. k by Th5,NAT_1:11;
  a ^ a <> x by A2,FLANG_1:11;
  hence thesis by A1,A4,A3,TARSKI:def 1;
end;
