reserve FT for non empty RelStr;
reserve x, y, z for Element of FT;
reserve A for Subset of FT;
reserve F for Subset of FT;

theorem
  x in A^s implies not x in (A \ {x})^b
proof
  assume x in A^s;
  then A misses (U_FT x \ {x}) by Th9;
  then A /\ (U_FT x \ {x}) = {};
  then
A1: (A /\ U_FT x) \ {x} = {} by XBOOLE_1:49;
  now
    per cases by A1,ZFMISC_1:58;
    suppose
      A /\ U_FT x = {};
      then A misses U_FT x;
      hence (A \ {x}) misses U_FT x by XBOOLE_1:36,63;
    end;
    suppose
      A /\ U_FT x = {x};
      then (U_FT x /\ A) \ {x} = {} by XBOOLE_1:37;
      then U_FT x /\ (A \ {x}) = {} by XBOOLE_1:49;
      hence (A \ {x}) misses U_FT x;
    end;
  end;
  hence thesis by Th8;
end;
