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
  F is discrete iff (for p ex O st p in O & INTERSECTION({O},F)\{{}} is
  trivial) & for A,B st A in F & B in F holds A=B or A misses B
proof
  now
    let F;
    (for p ex O st p in O & INTERSECTION({O},F)\{{}} is trivial) & (for A,
    B st A in F & B in F holds A=B or A misses B) implies F is discrete
    proof
      assume that
A1:   for p ex O st p in O & INTERSECTION({O},F)\{{}} is trivial and
A2:   for A,B st A in F & B in F holds A=B or A misses B;
      assume not F is discrete;
      then consider p such that
A3:   for O holds not p in O or not (for A,B st A in F & B in F holds
      O meets A & O meets B implies A=B);
      consider O such that
A4:   p in O and
A5:   INTERSECTION({O},F)\{{}} is trivial by A1;
      consider A,B such that
A6:   A in F and
A7:   B in F and
A8:   O meets A and
A9:   O meets B and
A10:  A<>B by A3,A4;
A11:  O/\B<>{} by A9,XBOOLE_0:def 7;
      set I=INTERSECTION({O},F);
      consider a being object such that
A12:  I\{{}}={} or I\{{}}={a} by A5,ZFMISC_1:131;
A13:  O in {O} by ZFMISC_1:31;
      then O/\B in I by A7,SETFAM_1:def 5;
      then O/\B in I\{{}} by A11,ZFMISC_1:56;
      then {O/\B} c= {a} by A12,ZFMISC_1:31;
      then
A14:  O/\B=a by ZFMISC_1:3;
A15:  O/\A<>{} by A8,XBOOLE_0:def 7;
      then consider f being object such that
A16:  f in O/\A by XBOOLE_0:def 1;
A17:  f in A by A16,XBOOLE_0:def 4;
      O/\A in I by A6,A13,SETFAM_1:def 5;
      then O/\A in I\{{}} by A15,ZFMISC_1:56;
      then {O/\A}c={a} by A12,ZFMISC_1:31;
      then O/\A=a by ZFMISC_1:3;
      then f in B by A14,A16,XBOOLE_0:def 4;
      then
A18:  f in A/\B by A17,XBOOLE_0:def 4;
      A misses B by A2,A6,A7,A10;
      hence thesis by A18,XBOOLE_0:def 7;
    end;
    hence
    F is discrete iff (for p ex O st p in O & INTERSECTION({O},F)\{{}} is
    trivial)& for A,B st A in F & B in F holds A=B or A misses B by Th6,Th7;
  end;
  hence thesis;
end;
