The left hand side of the equation is equal to the right hand side of it. Hence, proved.

Wouldn't it be grammatically like this-

The left hand side of the equation is equal to the right hand side of it. Hence, it is proved.

I hope the auxiliary verb are is used correctly in the given below another example.

The ongoing call or data session are not disconnected, hence no financial loss.

The ongoing call or data session are not disconnected, hence there is no financial loss.

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    It's definitely not "grammatical". Whether it's idiomatic or not is more a matter of opinion. In my opinion you should avoid things like this - it's really just throwing a couple of meaningful words out and leaving it to the audience to figure out how they combine to actually mean something. – FumbleFingers Reinstate Monica Aug 7 '17 at 17:37
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    I think your hence no financial loss is structurally akin to the "minimalist" (read: missing) modern syntax of, for example, I'm late because YouTube. Learners will often say things like this because they don't know any better - but the fact that some native speakers now do it deliberately isn't exactly a good reason for copying them. – FumbleFingers Reinstate Monica Aug 7 '17 at 17:42
  • Is your question specifically about sentences using the adverb "hence" Please clarify. Also "is" in your examples is the main verb of the clause, not an auxillary, again, please edit to correct this. – James K Aug 7 '17 at 18:52

The word hence is used to say to the listener, "and thus, what we are seeking to demonstrate has been demonstrated". It is already curt.

"Hence, there is no financial loss" and "hence it is proved" would be grammatical and normal. The ellipsis (hence, proved; hence no financial loss) would sound a little strange in a formal context if there were no clear and compelling reason for such brevity.

Such brevity is more commonly associated with a teacher in front of a blackboard going over the logical steps once again, say, or with a businessperson going through a list of items and not wanting to repeat the same formulaic words with each item visited.

  • Does the term ellipsis apply even when the subject part is omitted along with the verb? – Anubhav Singh Aug 8 '17 at 10:03
  • @Anubhav Singh: I believe I may not understand your question. What do you mean by "term" and what do you mean by "apply"? Are you intimating that it is possible to analyze the construction differently? – Tᴚoɯɐuo Aug 8 '17 at 11:43
  • Note that as with all parallel structures, repeated elements are liable to disappear without any laziness. "She is a genius and hence highly promotable" is neither informal nor formal. Are you thinking of this kind of thing, @AnubhavSingh?… – Luke Sawczak Aug 8 '17 at 13:28
  • Hence my question. – Tᴚoɯɐuo Aug 8 '17 at 13:37
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    @Anubhav Singh: Where are you getting this narrow definition of 'ellipsis'? – Tᴚoɯɐuo Aug 8 '17 at 16:59

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