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 Th10:
  <%>E in A |^.. n iff n = 0 or <%>E in A
proof
  thus <%>E in A |^.. n implies n = 0 or <%>E in A
  proof
    assume <%>E in A |^.. n;
    then
A1: ex k st n <= k & <%>E in A |^ k by Th2;
    n = 0 or n > 0;
    hence thesis by A1,FLANG_1:31;
  end;
  assume
A2: n = 0 or <%>E in A;
  per cases by A2;
  suppose
A3: n = 0;
    {<%>E} = A |^ 0 by FLANG_1:24;
    then <%>E in A |^ n by A3,TARSKI:def 1;
    hence thesis by Th2;
  end;
  suppose
    <%>E in A;
    then <%>E in A |^ n by FLANG_1:30;
    hence thesis by Th2;
  end;
end;
