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 Th26:
  for F being Subset-Family of GX st (for A being Subset of GX st
  A in F holds A is connected) & meet F <> {}GX 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: meet F <> {}GX;
  consider x being Point of GX such that
A3: x in meet F by A2,PRE_TOPC:12;
  meet F c= union F by SETFAM_1:2;
  then consider A2 being set such that
A4: x in A2 and
A5: A2 in F by A3,TARSKI:def 4;
  reconsider A2 as Subset of GX by A5;
A6: now
    let B be Subset of GX such that
A7: B in F and
    B <> A2;
    A2 c= Cl A2 by PRE_TOPC:18;
    then x in Cl A2 & x in B or x in A2 & x in Cl B by A3,A4,A7,SETFAM_1:def 1;
    then Cl A2 /\ B <> {} or A2 /\ Cl B <> {} by XBOOLE_0:def 4;
    then Cl A2 meets B or A2 meets Cl B;
    hence not A2,B are_separated;
  end;
  A2 <> {}GX by A4;
  hence thesis by A1,A5,A6,Th25;
end;
