theorem Th3:
  X is closed_wrt_A1-A7 implies {} in X
proof
  set o = the Element of X;
  reconsider p=o as Element of V;
  set D={{[0-element_of(V),x],[1-element_of(V),y]}: x in y & x in {p} & y in {
  p}};
A1: now
    set q = the Element of D;
    assume D <> {};
    then q in D;
    then consider x,y such that
    q={[0-element_of(V),x],[1-element_of(V),y]} and
A2: x in y and
A3: x in {p} & y in {p};
    x=p & y=p by A3,TARSKI:def 1;
    hence contradiction by A2;
  end;
  assume X is closed_wrt_A1-A7;
  then {p} in X & X is closed_wrt_A1 by Th2;
  hence thesis by A1;
end;
