reserve a,b,c,d for set,
  D,X1,X2,X3,X4 for non empty set,
  x1,y1,z1 for Element of X1,
  x2 for Element of X2,
  x3 for Element of X3,
  x4 for Element of X4,
  A1,B1 for Subset of X1;
reserve x,y for Element of [:X1,X2,X3:];
reserve x for Element of [:X1,X2,X3,X4:];
reserve A2 for Subset of X2,
  A3 for Subset of X3,
  A4 for Subset of X4;

theorem
  A1` = { x1 : not x1 in A1 }
proof
  thus A1` c= { x1 : not x1 in A1 }
  proof
    let a be object;
    assume
A1: a in A1`;
    then not a in A1 by XBOOLE_0:def 5;
    hence thesis by A1;
  end;
  let a be object;
  assume a in { x1 : not x1 in A1 };
  then ex x1 st a = x1 & not x1 in A1;
  hence thesis by SUBSET_1:29;
end;
