theorem
  X1,X2 are_separated implies X1 misses X2
proof
  reconsider A2 = the carrier of X2 as Subset of X by Th1;
  reconsider A1 = the carrier of X1 as Subset of X by Th1;
  assume X1,X2 are_separated;
  then A1,A2 are_separated;
  then A1 misses A2 by CONNSP_1:1;
  hence thesis;
end;
