reserve T, T1 for non empty TopSpace;
reserve F,G,H for Subset-Family of T,
  A,B,C,D for Subset of T,
  O,U for open Subset of T,
  p,q for Point of T,
  x,y,X for set;

theorem Th6:
  for F,A,B st F is discrete & A in F & B in F holds A=B or A misses B
proof
  let F,A,B;
  assume that
A1: F is discrete and
A2: A in F & B in F;
  assume that
A3: A<>B and
A4: A meets B;
  A/\B <> {}T by A4,XBOOLE_0:def 7;
  then consider p such that
A5: p in (A /\ B) by PRE_TOPC:12;
  consider O such that
A6: p in O and
A7: for C,D st C in F & D in F holds O meets C & O meets D implies C=D
  by A1;
A8: {p}c=O by A6,ZFMISC_1:31;
  p in B by A5,XBOOLE_0:def 4;
  then {p} c= B by ZFMISC_1:31;
  then {p} c=O/\B by A8,XBOOLE_1:19;
  then
A9: p in O/\B by ZFMISC_1:31;
  p in A by A5,XBOOLE_0:def 4;
  then {p}c=A by ZFMISC_1:31;
  then {p}c=O/\A by A8,XBOOLE_1:19;
  then
A10: p in O/\A by ZFMISC_1:31;
  O meets A & O meets B implies A=B by A2,A7;
  hence contradiction by A3,A10,A9,XBOOLE_0:4;
end;
