This first sentence:
1. The judge opined that the testimonial and documentary evidence in the record, before him, viewed in the fairest light, demonstrated against what has been averred, that which is being a claim by the defendant lady that circumstances and facts support a finding of mitigation under Penal Code Section 52-871(b), in respect of mistreatment as defined therein and as such conduct was directed to, and applied to, the victim, a young female.
-- Is grammatically "correct" for the most part. It has a long string of prepositional phrases and adjective clauses at the end of the independent clause to make it extremely long. It would sound better and would be more intelligible if it were broken up.
"that which is being a claim by the defendant lady"
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This part I do not understand in the least. After the comma are two dependent making words that indicate a relative clause coming up: 'that' and 'which'. Only one is necessary; two makes it redundant.
Furthermore, directly after this are two BE verbs right next to each other. One is a present tense BE verb "is," the other is a progressive participle "being". This implies that whatever "that" is, it is in constant application. Deleting "being" would clear it up.
Next, "defendant lady"? I have no idea what that means. It sounds like slang. Would it not be better to say "female defendant?"
There is a comma in front of "before him," I do not understand why: before him is in no way an interrupter, so get rid of it.
So, making the sentence look like this would be worlds better:
"The judge opined that the testimonial and documentary evidence in the record before him, viewed in the fairest light, demonstrated against what has been averred, which is a claim by the female defendant that...blah blah blah..."
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Ah...much better sounding. It might mean something a little different, so I might also suggest adding in another verb around the word "claim" and "defendant"...so that the defendant is actually making the claim. Since there is no verb, it is only implied in the writing.
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Now, on to the second sentence:
2. It having been viewed after the evidentiary hearing, and upon motion that the claim of mitigation by the stepmother be demurred upon in view of the evidence, it is found that cruelty was established by a preponderance of the evidence, in this being a civil case, and as such the claim of mitigation must be denied.
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This sentence is, as far as I can tell, correct. However, by using multiple progressive participles the author keeps the text going and makes it completely confusing. It would be much more efficient and intelligible to break it into two. For example:
"It having been viewed after the evidentiary hearing, and upon motion that the claim of mitigation by the stepmother be demurred upon in view of the evidence, it is found that cruelty was established by a preponderance of the evidence. This being a civil case, the claim of mitigation must be denied."
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Ah, I feel much better with that...whew. It sounds better; it is structurally more comfortable; AND it makes more sense by ridding the useless comma and deleting useless prepositional words.
Hope that helps.