reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;
reserve X for TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;

theorem
  for X1, X2 being closed non empty SubSpace of X holds X1 misses X2 iff
  X1,X2 are_separated
proof
  let X1, X2 be closed non empty SubSpace of X;
  reconsider A2 = the carrier of X2 as Subset of X by Th1;
  reconsider A1 = the carrier of X1 as Subset of X by Th1;
A1: A1 is closed & A2 is closed by Th11;
  thus X1 misses X2 implies X1,X2 are_separated
  by A1,Th34;
  assume X1,X2 are_separated;
  then A1,A2 are_separated;
  then A1 misses A2 by CONNSP_1:1;
  hence thesis;
end;
