
theorem
  for R,S being Relation, Y being set holds (R\/S)"Y = R"Y \/ S"Y
proof
  let R, S be Relation, Y be set;
  now
    let x be object;
    hereby
      assume x in (R \/ S)"Y;
      then consider y being object such that
        A1: [x,y] in R \/ S & y in Y by RELAT_1:def 14;
      per cases by A1, XBOOLE_0:def 3;
      suppose [x,y] in R;
        then x in R"Y by A1, RELAT_1:def 14;
        hence x in R"Y \/ S"Y by XBOOLE_0:def 3;
      end;
      suppose [x,y] in S;
        then x in S"Y by A1, RELAT_1:def 14;
        hence x in R"Y \/ S"Y by XBOOLE_0:def 3;
      end;
    end;
    assume x in R"Y \/ S"Y;
    then per cases by XBOOLE_0:def 3;
    suppose x in R"Y;
      then consider y being object such that
        A2: [x,y] in R & y in Y by RELAT_1:def 14;
      [x,y] in R \/ S by A2, XBOOLE_0:def 3;
      hence x in (R \/ S)"Y by A2, RELAT_1:def 14;
    end;
    suppose x in S"Y;
      then consider y being object such that
        A3: [x,y] in S & y in Y by RELAT_1:def 14;
      [x,y] in R \/ S by A3, XBOOLE_0:def 3;
      hence x in (R \/ S)"Y by A3, RELAT_1:def 14;
    end;
  end;
  hence thesis by TARSKI:2;
end;
