theorem
  W is closed iff ex x being set st W is_Walk_from x,x
proof
  hereby
    set x = W.first();
    assume W is closed;
    then W.first() = W.last();
    then W is_Walk_from x,x;
    hence ex x being set st W is_Walk_from x,x;
  end;
  given v being set such that
A1: W is_Walk_from v,v;
A2: W.last() = v by A1;
  W.first() = v by A1;
  hence thesis by A2;
end;
