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

theorem Th3:
  {_{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;
    x in X or x in Y by A2,XBOOLE_0:def 3;
    then y in {_{X}_} or y in {_{Y}_} by A1,Th1;
    hence thesis by XBOOLE_0:def 3;
  end;
  let y be object;
  assume
A3: y in {_{X}_} \/ {_{Y}_};
  per cases by A3,XBOOLE_0:def 3;
  suppose y in {_{X}_};
    then consider x being object such that
A4: y = {x} and
A5: x in X by Th1;
    x in X\/Y by A5,XBOOLE_0:def 3;
    hence thesis by A4,Th1;
  end;
  suppose y in {_{Y}_};
    then consider x being object such that
A6: y = {x} and
A7: x in Y by Th1;
    x in X\/Y by A7,XBOOLE_0:def 3;
    hence thesis by A6,Th1;
  end;
end;
