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 Th46:
  <%x%> in A |^.. k iff <%x%> in A & (<%>E in A or k <= 1)
proof
  thus <%x%> in A |^.. k implies <%x%> in A & (<%>E in A or k <= 1)
  proof
    assume <%x%> in A |^.. k;
    then ex m st k <= m & <%x%> in A |^ m by Th2;
    hence thesis by FLANG_2:9;
  end;
  assume that
A1: <%x%> in A and
A2: <%>E in A or k <= 1;
  per cases by A2,NAT_1:25;
  suppose
    <%>E in A & k > 1;
    then <%x%> in A |^ k by A1,FLANG_2:9;
    hence thesis by Th2;
  end;
  suppose
    k = 0;
    then A |^.. k = A* by Th11;
    hence thesis by A1,FLANG_1:72;
  end;
  suppose
    k = 1;
    then <%x%> in A |^ k by A1,FLANG_1:25;
    hence thesis by Th2;
  end;
end;
