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 \ B1 = { x1 : x1 in A1 & not x1 in B1 }
proof
  thus A1 \ B1 c= { x1 : x1 in A1 & not x1 in B1 }
  proof
    let a be object;
    assume
A1: a in A1 \ B1;
    then reconsider x = a as Element of X1;
    x in A1 & not x in B1 by A1,XBOOLE_0:def 5;
    hence thesis;
  end;
  let a be object;
  assume a in { x1 : x1 in A1 & not x1 in B1 };
  then ex x1 st a = x1 & x1 in A1 & not x1 in B1;
  hence thesis by XBOOLE_0:def 5;
end;
