reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;

theorem
  SFY <> {} implies X \ union SFY = meet DIFFERENCE({X},SFY)
proof
  set y = the Element of SFY;
A1: X in {X} by TARSKI:def 1;
  assume SFY <> {};
  then
A2: X \ y in DIFFERENCE({X},SFY) by A1,Def6;
A3: meet DIFFERENCE({X},SFY) c= X \ union SFY
  proof
    let x be object;
    assume
A4: x in meet DIFFERENCE({X},SFY);
    for Z st Z in SFY holds not x in Z
    proof
      let Z;
      assume Z in SFY;
      then X \ Z in DIFFERENCE({X},SFY) by A1,Def6;
      then x in X \ Z by A4,Def1;
      hence thesis by XBOOLE_0:def 5;
    end;
    then for Z st x in Z holds not Z in SFY;
    then
A5: not x in union SFY by TARSKI:def 4;
    x in X \ y by A2,A4,Def1;
    hence thesis by A5,XBOOLE_0:def 5;
  end;
  X \ union SFY c= meet DIFFERENCE({X},SFY)
  proof
    let x be object;
    assume x in X \ union SFY;
    then
A6: x in X & not x in union SFY by XBOOLE_0:def 5;
    for Z st Z in DIFFERENCE({X},SFY) holds x in Z
    proof
      let Z;
      assume Z in DIFFERENCE({X},SFY);
      then consider Z1,Z2 such that
A7:   Z1 in {X} & Z2 in SFY and
A8:   Z = Z1 \ Z2 by Def6;
      x in Z1 & not x in Z2 by A6,A7,TARSKI:def 1,def 4;
      hence thesis by A8,XBOOLE_0:def 5;
    end;
    hence thesis by A2,Def1;
  end;
  hence thesis by A3;
end;
