
theorem Th17:
  for A being non empty Interval, x being Real holds x ** A is
  Interval
proof
  let A be non empty Interval;
  let x be Real;
  per cases;
  suppose
    x = 0;
    hence thesis by Th6;
  end;
  suppose
A1: x <> 0;
    now
      per cases by MEASURE5:1;
      case
        A is open_interval;
        then x ** A is open_interval by A1,Th7;
        hence thesis;
      end;
      case
        A is closed_interval;
        then x ** A is closed_interval by A1,Th8;
        hence thesis;
      end;
      case
A2:     A is right_open_interval;
        per cases by A1;
        case
          x < 0;
          then x ** A is left_open_interval by A2,Th10;
          hence thesis;
        end;
        case
          0 < x;
          then x ** A is right_open_interval by A2,Th9;
          hence thesis;
        end;
      end;
      case
A3:     A is left_open_interval;
        now
          per cases by A1;
          case
            x < 0;
            then x ** A is right_open_interval by A3,Th12;
            hence thesis;
          end;
          case
            0 < x;
            then x ** A is left_open_interval by A3,Th11;
            hence thesis;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
end;
