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 Th51:
  m > 0 & A |^ (m, n) = {x} implies for mn st m <= mn & mn <= n
  holds A |^ mn = {x}
proof
  assume that
A1: m > 0 and
A2: A |^ (m, n) = {x};
  given mn such that
A3: m <= mn & mn <= n and
A4: A |^ mn <> {x};
  per cases;
  suppose
A5: A |^ mn = {};
    x in A |^ (m, n) by A2,TARSKI:def 1;
    then
A6: ex i st m <= i & i <= n & x in A |^ i by Th19;
    A = {} by A5,FLANG_1:27;
    hence contradiction by A1,A6,FLANG_1:27;
  end;
  suppose
    A |^ mn <> {};
    then consider y being object such that
A7: y in A |^ mn and
A8: y <> x by A4,ZFMISC_1:35;
    y in A |^ (m, n) by A3,A7,Th19;
    hence contradiction by A2,A8,TARSKI:def 1;
  end;
end;
