
theorem
  for R,S being Relation, X being set holds (R\/S).:X = R.:X \/ S.:X
proof
  let R, S be Relation, X be set;
  now
    let y be object;
    hereby
      assume y in (R \/ S).:X;
      then consider x being object such that
        A1: [x,y] in R \/ S & x in X by RELAT_1:def 13;
      per cases by A1, XBOOLE_0:def 3;
      suppose [x,y] in R;
        then y in R.:X by A1, RELAT_1:def 13;
        hence y in R.:X \/ S.:X by XBOOLE_0:def 3;
      end;
      suppose [x,y] in S;
        then y in S.:X by A1, RELAT_1:def 13;
        hence y in R.:X \/ S.:X by XBOOLE_0:def 3;
      end;
    end;
    assume y in R.:X \/ S.:X;
    then per cases by XBOOLE_0:def 3;
    suppose y in R.:X;
      then consider x being object such that
        A2: [x,y] in R & x in X by RELAT_1:def 13;
      [x,y] in R \/ S by A2, XBOOLE_0:def 3;
      hence y in (R \/ S).:X by A2, RELAT_1:def 13;
    end;
    suppose y in S.:X;
      then consider x being object such that
        A3: [x,y] in S & x in X by RELAT_1:def 13;
      [x,y] in R \/ S by A3, XBOOLE_0:def 3;
      hence y in (R \/ S).:X by A3, RELAT_1:def 13;
    end;
  end;
  hence thesis by TARSKI:2;
end;
