reserve a,b,c for set;

theorem Th13:
  for T being discrete non empty TopStruct holds SmallestPartition
the carrier of T is Basis of T & for B being Basis of T holds SmallestPartition
  the carrier of T c= B
proof
  let T be discrete non empty TopStruct;
  set B0 = SmallestPartition the carrier of T;
A1: B0 c= the topology of T
  proof
    let a be object;
    assume a in B0;
    then reconsider a as Subset of T;
    a is open by TDLAT_3:15;
    hence thesis;
  end;
A2: B0 = the set of all {a} where a is Element of T by EQREL_1:37;
  now
    let A be Subset of T such that
    A is open;
    let p be Point of T such that
A3: p in A;
    reconsider a = {p} as Subset of T;
    take a;
    thus a in B0 by A2;
    thus p in a by TARSKI:def 1;
    thus a c= A by A3,ZFMISC_1:31;
  end;
  hence
A4: B0 is Basis of T by A1,YELLOW_9:32;
  let B be Basis of T;
  let a be object;
  assume
A5: a in B0;
  then consider b being Element of T such that
A6: a = {b} by A2;
  reconsider a as Subset of T by A5;
A7: b in a by A6,TARSKI:def 1;
  a is open by A4,A5,YELLOW_8:10;
  then consider U being Subset of T such that
A8: U in B and
A9: b in U and
A10: U c= a by A7,YELLOW_9:31;
  a c= U by A6,A9,ZFMISC_1:31;
  hence thesis by A8,A10,XBOOLE_0:def 10;
end;
