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

theorem
  SFY <> {} implies X \ meet SFY = union DIFFERENCE ({X},SFY)
proof
A1: X in {X} by TARSKI:def 1;
A2: union DIFFERENCE({X},SFY) c= X \ meet SFY
  proof
    let x be object;
    assume x in union DIFFERENCE({X},SFY);
    then consider Z such that
A3: x in Z and
A4: Z in DIFFERENCE({X},SFY) by TARSKI:def 4;
    consider Z1,Z2 such that
A5: Z1 in {X} and
A6: Z2 in SFY and
A7: Z = Z1 \ Z2 by A4,Def6;
    not x in Z2 by A3,A7,XBOOLE_0:def 5;
    then
A8: not x in meet SFY by A6,Def1;
    Z1 = X by A5,TARSKI:def 1;
    hence thesis by A3,A7,A8,XBOOLE_0:def 5;
  end;
  assume
A9: SFY <> {};
  X \ meet SFY c= union DIFFERENCE({X},SFY)
  proof
    let x be object;
    assume
A10: x in X \ meet SFY;
    then not x in meet SFY by XBOOLE_0:def 5;
    then consider Z such that
A11: Z in SFY and
A12: not x in Z by A9,Def1;
A13: x in X \ Z by A10,A12,XBOOLE_0:def 5;
    X \ Z in DIFFERENCE({X},SFY) by A1,A11,Def6;
    hence thesis by A13,TARSKI:def 4;
  end;
  hence thesis by A2;
end;
