reserve a,b,c for set;

theorem
  for X being non empty set, x0 being set holds ex B0 being prebasis of
DiscrWithInfin(X,x0) st B0 = ((SmallestPartition X) \ {{x0}}) \/ the set of
all {x}` where x
  is Element of X
proof
  let X be non empty set;
  let x0 be set;
  set T = DiscrWithInfin(X,x0);
  set SX = SmallestPartition X, FX = {F` where F is Subset of X: F is finite},
  AX = the set of all {x}` where x is Element of X;
  reconsider SX as Subset-Family of T by Def5;
  AX c= bool X
  proof
    let a be object;
    assume a in AX;
    then ex x being Element of X st a = {x}`;
    hence thesis;
  end;
  then reconsider AX as Subset-Family of T by Def5;
  reconsider pB = (SX\{{x0}})\/AX as Subset-Family of T;
  consider B0 being Basis of T such that
A1: B0 = ((SmallestPartition X) \ {{x0}}) \/ FX by Th29;
A2: the carrier of T = X by Def5;
A3: FX c= FinMeetCl pB
  proof
    let a be object;
    assume a in FX;
    then consider F being Subset of T such that
A4: a = F` and
A5: F is finite by A2;
    set SF = SmallestPartition F;
    bool F c= bool X by A2,ZFMISC_1:67;
    then reconsider SF as Subset-Family of T by A2,XBOOLE_1:1;
    per cases;
    suppose
      F = {};
      hence thesis by A4,CANTOR_1:8;
    end;
    suppose
      F <> {};
      then reconsider F as non empty Subset of T;
      SF is a_partition of F;
      then reconsider SF as non empty Subset-Family of T;
A6:   COMPLEMENT SF c= pB
      proof
        let a be object;
        assume
A7:     a in COMPLEMENT SF;
        then reconsider a as Subset of T;
        a` in SF by A7,SETFAM_1:def 7;
        then a` in the set of all {b} where b is Element of F by
EQREL_1:37;
        then consider b being Element of F such that
A8:     a` = {b};
        reconsider b as Element of X by Def5;
        {b}` in AX;
        hence thesis by A2,A8,XBOOLE_0:def 3;
      end;
      F = union SF by EQREL_1:def 4;
      then
A9:   a = meet COMPLEMENT SF by A4,TOPS_2:6;
      COMPLEMENT SF is finite by A5,TOPS_2:8;
      then Intersect COMPLEMENT SF in FinMeetCl pB by A6,CANTOR_1:def 3;
      hence thesis by A9,SETFAM_1:def 9;
    end;
  end;
A10: pB c= FinMeetCl pB by CANTOR_1:4;
A11: B0 c= FinMeetCl pB
  proof
    let a be object;
    assume a in B0;
    then a in SX\{{x0}} or a in FX by A1,XBOOLE_0:def 3;
    then a in pB or a in FX by XBOOLE_0:def 3;
    hence thesis by A10,A3;
  end;
A12: B0 c= the topology of T by TOPS_2:64;
  AX c= FX
  proof
    let a be object;
    assume a in AX;
    then ex x being Element of X st a = {x}`;
    hence thesis;
  end;
  then pB c= B0 by A1,XBOOLE_1:9;
  then pB c= the topology of T by A12;
  then FinMeetCl pB c= FinMeetCl the topology of T by CANTOR_1:14;
  then FinMeetCl pB c= the topology of T by CANTOR_1:5;
  then FinMeetCl pB is Basis of T by A11,WAYBEL19:22;
  then pB is prebasis of T by YELLOW_9:23;
  hence thesis;
end;
