reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem
  {_{X\Y}_} = {_{X}_} \ {_{Y}_}
proof
  thus {_{X\Y}_} c= {_{X}_} \ {_{Y}_}
  proof
    let y be object;
    assume y in {_{X\Y}_};
    then consider x being object such that
A1: y = {x} and
A2: x in X\Y by Th1;
A3: not x in Y by A2,XBOOLE_0:def 5;
A4: y in {_{X}_} by A1,A2,Th1;
    not y in {_{Y}_}
    proof
      assume not thesis;
      then ex x1 being object st ( y = {x1})&( x1 in Y) by Th1;
      hence contradiction by A1,A3,ZFMISC_1:3;
    end;
    hence thesis by A4,XBOOLE_0:def 5;
  end;
  let y be object;
  assume
A5: y in {_{X}_} \ {_{Y}_};
  then
A6: y in {_{X}_};
A7: not y in {_{Y}_} by A5,XBOOLE_0:def 5;
  consider x being object such that
A8: y = {x} and
A9: x in X by A6,Th1;
  not x in Y by A7,A8,Th1;
  then x in X\Y by A9,XBOOLE_0:def 5;
  hence thesis by A8,Th1;
end;
