
theorem Th93:
  for L being non empty doubleLoopStr, F being non empty Subset of
  L st F <> {0.L} ex x being Element of L st x <> 0.L & x in F
proof
  let L be non empty doubleLoopStr, F be non empty Subset of L;
  assume
A1: F <> {0.L};
  now
    assume
A2: for x being set st x in F holds x = 0.L;
    for x being object holds x in F iff x = 0.L
    proof
      let e be object;
A3:   ex a being object st a in F by XBOOLE_0:def 1;
      thus e in F implies e = 0.L by A2;
      assume e = 0.L;
      hence thesis by A2,A3;
    end;
    hence contradiction by A1,TARSKI:def 1;
  end;
  hence thesis;
end;
