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 Th28:
  for GX being non empty TopSpace st ex x being Point of GX st for
  y being Point of GX holds x,y are_joined holds GX is connected
proof
  let GX be non empty TopSpace;
  given a being Point of GX such that
A1: for x being Point of GX holds a,x are_joined;
  now
    let x be Point of GX;
    defpred P[set] means ex C1 being Subset of GX st C1 = $1 & C1 is connected
    & x in $1;
    consider F being Subset-Family of GX such that
A2: for C being Subset of GX holds C in F iff P[C] from SUBSET_1:sch 3;
    take F;
    let C be Subset of GX;
    thus C in F implies C is connected & x in C
    proof
      assume C in F;
      then
      ex C1 being Subset of GX st C1 = C & C1 is connected & x in C by A2;
      hence thesis;
    end;
    thus C is connected & x in C implies C in F by A2;
  end;
  then consider F being Subset-Family of GX such that
A3: for C being Subset of GX holds C in F iff C is connected & a in C;
A4: for x being Point of GX
      ex C being Subset of GX st C is connected & a in C & x in C by A1,Def4;
  now
    let x be object;
    assume x in [#]GX;
    then consider C being Subset of GX such that
A5: C is connected and
A6: a in C and
A7: x in C by A4;
    C in F by A3,A5,A6;
    hence x in union F by A7,TARSKI:def 4;
  end;
  then [#]GX c= union F by TARSKI:def 3;
  then
A8: union F = [#]GX;
A9: for A being set st A in F holds a in A by A3;
    a in {a} by TARSKI:def 1;
  then  F <> {} by A3;
  then
A10: meet F <> {}GX by A9,SETFAM_1:def 1;
  for A being Subset of GX st A in F holds A is connected by A3;
  then [#]GX is connected by A8,A10,Th26;
  hence thesis by Th27;
end;
