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 Th10:
  m <> n & A |^ m = {x} & A |^ n = {x} implies x = <%>E
proof
  assume that
A1: m <> n and
A2: A |^ m = {x} and
A3: A |^ n = {x};
A4: x in A |^ m by A2,TARSKI:def 1;
A5: x in A |^ n by A3,TARSKI:def 1;
  per cases by A1,XXREAL_0:1;
  suppose
    m > n;
    then consider k such that
A6: n + k = m and
A7: k > 0 by Th3;
    (A |^ n) ^^ (A |^ k) = A |^ m by A6,FLANG_1:33;
    then consider a, b such that
A8: a in A |^ n and
A9: b in A |^ k and
A10: x = a ^ b by A4,FLANG_1:def 1;
    a = x by A3,A8,TARSKI:def 1;
    then b = <%>E by A10,Th4;
    then <%>E in A by A7,A9,FLANG_1:31;
    then <%>E in A |^ m by FLANG_1:30;
    hence thesis by A2;
  end;
  suppose
    m < n;
    then consider k such that
A11: m + k = n and
A12: k > 0 by Th3;
    (A |^ m) ^^ (A |^ k) = A |^ n by A11,FLANG_1:33;
    then consider a, b such that
A13: a in A |^ m and
A14: b in A |^ k and
A15: x = a ^ b by A5,FLANG_1:def 1;
    a = x by A2,A13,TARSKI:def 1;
    then b = <%>E by A15,Th4;
    then <%>E in A by A12,A14,FLANG_1:31;
    then <%>E in A |^ m by FLANG_1:30;
    hence thesis by A2;
  end;
end;
