In my paper, I have used the expression "It is well known that" twice in a row. Using the same expression repeatedly may appear monotonous. Are there any other commonly used alternatives for this expression?
It is well known that every planar graph with $n\,(\ge3)$ vertices has at most $3n-6$ edges, and thus has the minimum degree at most 5. It is known that any planar graph with minimum degree 5 has at least 12 vertices and any planar graph of odd order with minimum degree 5 has at least 15.
I attempted to ask GPT, and its responses seem uncommon in mathematical research papers.
It suggests I use the following sentences:
"It is a known fact that..."
"It is widely recognized that..."
"It is an established result that..."
"It is commonly understood that..."
"It is widely accepted that..."
Note:
Yes, this question is better suited for English Stack Exchange. It should be deleted, but after seeing the nice suggestions provided by Will Jagy, I decided to keep it.
By the way, the suggestions of Xander Henderson and Greg Martin are very helpful; citing references is better than using "It is well known."
It is well known that every planar graph with $n\,(\ge3)$ vertices has at most $3n-6$ edges, therefore having minimum degree at most 5. A planar graph with minimum degree 5 has at least 12 vertices. Finally, a planar graph of odd order, with minimum degree 5, has at least 15 vertices.