reserve GX for TopSpace;
reserve A, B, C for Subset of GX;
reserve TS for TopStruct;
reserve K, K1, L, L1 for Subset of TS;

theorem Th25:
  for F being Subset-Family of GX st (for A being Subset of GX st
A in F holds A is connected) & (ex A being Subset of GX st A <> {}GX & A in F &
  (for B being Subset of GX st B in F & B <> A holds not A,B are_separated))
  holds union F is connected
proof
  let F be Subset-Family of GX;
  assume that
A1: for A being Subset of GX st A in F holds A is connected and
A2: ex A being Subset of GX st A <> {}GX & A in F & for B being Subset
  of GX st B in F & B <> A holds not A,B are_separated;
  consider A1 being Subset of GX such that
  A1 <> {}GX and
A3: A1 in F and
A4: for B being Subset of GX st B in F & B <> A1 holds not A1,B
  are_separated by A2;
  reconsider A1 as Subset of GX;
  now
    let P,Q be Subset of GX;
    assume that
A5: union F = P \/ Q and
A6: P,Q are_separated;
    P misses Q by A6,Th1;
    then
A7: P /\ Q = {};
A8: now
      assume
A9:   A1 c= Q;
      now
        let B be Subset of GX;
        assume that
A10:    B in F and
        B <> A1 and
A11:    not B c= Q;
        B is connected by A1,A10;
        then B c= P or B c= Q by A5,A6,A10,Th16,ZFMISC_1:74;
        hence contradiction by A4,A6,A9,A10,A11,Th7;
      end;
      then for A being set st A in F holds A c= Q by A9;
      then
A12:  union F c= Q by ZFMISC_1:76;
      Q c= P \/ Q by XBOOLE_1:7;
      then Q = P \/ Q by A5,A12;
      hence P = {}GX by A7,XBOOLE_1:7,28;
    end;
A13: now
      assume
A14:  A1 c= P;
      now
        let B be Subset of GX;
        assume that
A15:    B in F and
        B <> A1;
        B is connected by A1,A15;
        then
A16:    B c= P or B c= Q by A5,A6,A15,Th16,ZFMISC_1:74;
        assume not B c= P;
        hence contradiction by A4,A6,A14,A15,A16,Th7;
      end;
      then for A being set st A in F holds A c= P by A14;
      then
A17:  union F c= P by ZFMISC_1:76;
      P c= P \/ Q by XBOOLE_1:7;
      then P = P \/ Q by A5,A17;
      hence Q = {}GX by A7,XBOOLE_1:7,28;
    end;
    A1 c= P \/ Q by A3,A5,ZFMISC_1:74;
    hence P = {}GX or Q = {}GX by A1,A3,A6,A13,A8,Th16;
  end;
  hence thesis by Th15;
end;
