Definition of "in fact"

Now all this was very fine, but not at all in keeping with the Celebrity's character as I had come to conceive it. The idea that adulation ever cloyed on him was ludicrous in itself. In fact I thought the whole story fishy, and came very near to saying so.

1897 December (indicated as 1898), Winston Churchill, chapter III, in The Celebrity: An Episode, New York, N.Y.: The Macmillan Company; London: Macmillan & Co., Ltd.

A great bargain also had been the excellent Axminster carpet which covered the floor; as, again, the arm-chair in which Bunting now sat forward, staring into the dull, small fire. In fact, that arm-chair had been an extravagance of Mrs. Bunting. She had wanted her husband to be comfortable after the day's work was done, and she had paid thirty-seven shillings for the chair.

1913, Mrs. [Marie] Belloc Lowndes, chapter I, in The Lodger, London: Methuen; republished in Novels of Mystery: The Lodger; The Story of Ivy; What Really Happened, New York, N.Y.: Longmans, Green and Co., […], page 0016

Menggetimur (d. 1433),¹⁸ chief of the Odoli subtribe of the Chien-chou Jurchen, had in fact moved into Korea south of the Tumen River. Because he offered tribute to the Choson court and had encamped at Hoeryŏng (Chinese: Hui-ning), the Koreans regarded him as their vassal.

1998, Donald N. Clark, “Sino-Korean tributary relations under the Ming”, in Denis Twitchett, Frederick W. Mote, editors, The Cambridge History of China‎, volume 8, number 2, Cambridge University Press, page 286

We tend to think of the seafloor a few kilometres down as a flat plain. In fact, about two-thirds of this “abyssal” seabed is made up of gentle rolling hills a few hundred metres high, says Jennifer Durden at the University of Southampton, UK.

2015 August 8, Bob Holmes, “Ocean hills yield secret ecosystems”, in New Scientist‎, number 3033, archived from the original on 5 September 2015, page 14