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 Th52:
  m <> n & A |^ (m, n) = {x} implies x = <%>E
proof
  assume that
A1: m <> n and
A2: A |^ (m, n) = {x};
  per cases;
  suppose
    m > n;
    hence thesis by A2,Th21;
  end;
  suppose
A3: m = 0 & m <= n;
    then {<%>E} = A |^ m by FLANG_1:24;
    then <%>E in A |^ m by TARSKI:def 1;
    then <%>E in A |^ (m, n) by A3,Th19;
    hence thesis by A2;
  end;
  suppose
    m > 0 & m <= n;
    then A |^ m = {x} & A |^ n = {x} by A2,Th51;
    hence thesis by A1,Th10;
  end;
end;
