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 Th31:
  for GX being non empty TopSpace for x being Point of GX, F being
Subset-Family of GX st for A being Subset of GX holds A in F iff A is connected
  & x in A holds F <> {}
proof
  let GX be non empty TopSpace;
  let x be Point of GX, F be Subset-Family of GX such that
A1: for A being Subset of GX holds A in F iff A is connected & x in A;
  x in {x} by TARSKI:def 1;
  hence thesis by A1;
end;
