reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;

theorem Th11:
  {x} = meet MaxADSF(x)
proof
A1: x in {x} by TARSKI:def 1;
  now
    let A be set;
    assume A in MaxADSF(x);
    then ex C being Subset of Y st C = A & C is anti-discrete & x in C;
    hence {x} c= A by ZFMISC_1:31;
  end;
  then
A2: {x} c= meet MaxADSF(x) by SETFAM_1:5;
  {x} is anti-discrete by Th6;
  then {x} in MaxADSF(x) by A1;
  then meet MaxADSF(x) c= {x} by SETFAM_1:3;
  hence thesis by A2;
end;
