reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;

theorem Th2:
  a in b & c in d implies
  c <> a & c <> b & d <> a & d <> b     or
  c in a & d = a   or   c in a & d = b  or
  c = a & d in b   or   c = a & d = b   or   c = a & b in d  or
  a in c & d = b   or   c = b & b in d
  proof assume
A1: a in b & c in d;
    per cases by ORDINAL1:14;
    suppose
      c in a;
      hence thesis by A1;
    end;
    suppose
      c = a;
      hence thesis by ORDINAL1:14;
    end;
    suppose
A2:   a in c & c in b;
      per cases by ORDINAL1:14;
      suppose d in b;
        hence thesis by A1;
      end;
      suppose d = b;
        hence thesis by A2;
      end;
      suppose b in d;
        hence thesis by A1;
      end;
    end;
    suppose
      c = b;
      hence thesis by A1;
    end;
    suppose
      b in c;
      hence thesis by A1;
    end;
  end;
