theorem Th1:
  X in PA & V in PA & X c= V implies X=V
proof
  assume that
A1: X in PA and
A2: V in PA and
A3: X c= V;
A4: X=V or X misses V by A1,A2,EQREL_1:def 4;
  set x = the Element of X;
  X<>{} by A1,EQREL_1:def 4;
  then x in X;
  hence thesis by A3,A4,XBOOLE_0:3;
end;
