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The expression "on the contrary though" is correct and can be used in written English
You can use it when you want to express an idea that is contrary to a previously mentioned idea. For example, you could say: "I thought the project was too difficult to complete in that time frame, but on the contrary though, we managed to finish it in three weeks."
Exact(6)
On the contrary: though the Concertacion won December's congressional election comfortably, its share of the vote fell by 5%.
On the contrary, though the building itself is modest in dimensions, the eye has an opportunity to roam.
On the contrary, though older than most noblemen starting on the grand tour, he resolved to complete his education by foreign travel.
Such novels seldom try to startle or provoke the reader; on the contrary, though the narrative may unfold ingeniously, every art is employed to make the reader at ease within it.
On the contrary, though I disagreed with almost every student I talked to in Trafalgar Square and later at UCL in Bloomsbury (now in its seventh day of a sit-in), I found myself oddly relieved.The contrast was striking with student demonstrations I have reported on elsewhere, over the years.
On the contrary, though subjects with sTfR concentrations below the threshold represented the preponderance of women with normal vaginal microflora, a substantial proportion of the women with disturbed vaginal microflora also had no evidence of iron deficiency, and therefore the sensitivity and the PPV were actually low.
Similar(53)
On the contrary, sad though I shall be to leave behind those I love, I suspect the end of life, whatever happens, will be a relief.
On the contrary, even though the antigens could be brought into DCs by γFe2O3/DMSA, they failed to induce an effective T cell response (Fig. 2).
On the contrary, even though she often complained to OSAPE, she felt that this office could not really fight for her rights.
On the contrary, even though skilled agricultural workers (ISCO 6) report very frequent exposure to known physical risk factors (see polar plots in Figure 1), the corresponding odds ratios reported in Table 3 suggest that the differences between workers in ISCO group 1 and ISCO group 6 are accounted for by the variables specified in the regression models.
On the contrary, for the US (though not for countries like Spain), wage stickiness is if anything good for us right now, helping stave off destructive deflation.
More suggestions(15)
on the environment though
on the left though
on the airplane though
on the money though
on the program though
on the screen though
on the implementation though
on the phone though
on the hour though
on the case though
on the shank though
on the mother though
on the background though
on the field though
on the back though
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com