reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  R.:(X \/ Y) = R.:X \/ R.:Y
  proof
    thus R.:(X \/ Y) c= R.:X \/ R.:Y
    proof
      let y be object;
      assume y in R.:(X \/ Y);
      then consider x such that
A1:   [x,y] in R and
A2:   x in X \/ Y by Def11;
      x in X or x in Y by A2,XBOOLE_0:def 3;
      then y in R.:X or y in R.:Y by A1,Def11;
      hence y in R.:X \/ R.:Y by XBOOLE_0:def 3;
    end;
    let y be object;
    assume
A3: y in R.:X \/ R.:Y;
    per cases by A3,XBOOLE_0:def 3;
    suppose y in R.:Y;
      then consider x such that
A4:   [x,y] in R and
A5:   x in Y by Def11;
      x in X \/ Y by A5,XBOOLE_0:def 3;
      hence y in R.:(X \/ Y) by A4,Def11;
    end;
    suppose y in R.:X;
      then consider x such that
A6:   [x,y] in R and
A7:   x in X by Def11;
      x in X \/ Y by A7,XBOOLE_0:def 3;
      hence y in R.:(X \/ Y) by A6,Def11;
    end;
  end;
