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

theorem Th53:
  <%x%> in A |^ (m, n) iff <%x%> in A & m <= n & ( <%>E in A & n >
  0 or m <= 1 & 1 <= n )
proof
  thus <%x%> in A |^ (m, n) implies <%x%> in A & m <= n & ( <%>E in A & n > 0
  or m <= 1 & 1 <= n )
  proof
    assume
A1: <%x%> in A |^ (m, n);
    then ex mn st m <= mn & mn <= n & <%x%> in A |^ mn by Th19;
    hence thesis by A1,Th9,Th21;
  end;
  assume that
A2: <%x%> in A and
A3: m <= n and
A4: <%>E in A & n > 0 or m <= 1 & 1 <= n;
  per cases by A4;
  suppose
    <%>E in A & n > 0;
    then A c= A |^ n by FLANG_1:35;
    hence thesis by A2,A3,Th19;
  end;
  suppose
A5: m <= 1 & 1 <= n;
    <%x%> in A |^ 1 by A2,FLANG_1:25;
    hence thesis by A5,Th19;
  end;
end;
