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

theorem Th4:
  {_{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: x in X by A2,XBOOLE_0:def 4;
A4: x in Y by A2,XBOOLE_0:def 4;
A5: y in {_{X}_} by A1,A3,Th1;
    y in {_{Y}_} by A1,A4,Th1;
    hence thesis by A5,XBOOLE_0:def 4;
  end;
  let y be object;
  assume
A6: y in {_{X}_} /\ {_{Y}_};
  then
A7: y in {_{X}_} by XBOOLE_0:def 4;
A8: y in {_{Y}_} by A6,XBOOLE_0:def 4;
  consider x being object such that
A9: y = {x} and
A10: x in X by A7,Th1;
  consider x1 being object such that
A11: y = {x1} and
A12: x1 in Y by A8,Th1;
  x = x1 by A9,A11,ZFMISC_1:3;
  then x in X /\ Y by A10,A12,XBOOLE_0:def 4;
  hence thesis by A9,Th1;
end;
