memes

Property, not Law, yes

• an old textbook (p78. Note, before quibbling about "products", that even though it is expressed by juxtaposition, on page 1 this textbook says that, "multiplication is indicated by the absence of a sign" and we are in the multiplication, not the non-existent "product" section. The following exercises are explicitly multiplication.)
• a new textbook (p31) (Note that this time the law is expressed with × symbols. I also note that this textbook is the exact same publisher - CGP - that you have screenshotted in support of your own claims elsewhere, and indeed we used CGP textbooks at school.)
• and a teacher resource (p5) (published by the University of Melbourne for teaching primary and secondary students)

All call it the "distributive law". The second textbook uses the term interchangeably with "distributive property" (p34). A five-second google can find numerous webpages, with an introductory paragraph which starts, "the distributive law, also called the distributive property..." and Encylopedia Britannica says too that they are the same thing. All three resources make no distinction as far as distribution goes between an expression written with a multiplication symbol like 2×(4+5) and one without like 2(x+3).

There is no difference between the two terms. I'd ask you for your reference, but I'm sure it's in the same place as your references to your other nonsense that you tried and failed to find references for, so you needn't bother: the evidence above is all that's needed to dismiss this bollocks anyway. So stop repeating this stupid claim; nobody except a complete moron is going to use these two terms - which certainly sound like synonyms - to refer to two closely related, but different things. It's asking for confusion. I don't intend to discuss this point again.

who proved them, yes
from proof of same

So, you're saying that, some time after 1928, a mathematician proved that a ÷ b × c = (a ÷ b) × c? Where was this result published? What's the citation? Who was the author (or authors)? Or maybe you don't have the citation to hand, but know the proof off the top of your head? Please, let me see it.

Does it not seem weird to you that such a basic aspect of mathematics as multiplication and division remained undiscovered until the 20th century? It's hardly Fermat's Last Theorem. If the meaning of a ÷ b × c were not a matter of choice but instead an open question, why were mathematicians using the notation at all when its meaning was not known?

And what of the textbooks like High School Algebra, Elementary Course (1917) which used the convention - err oops, rule - of performing multiplication first? See page 212, example 155. (Would be 810, not 90, if using strict left-to-right priority for division and multiplication)

If it were proved that this ~~convention~~ is wrong, then you will surely be able to find some serious error that flows from doing division in this order. After all, from any contradiction, you can prove anything.

Of course you won't be able to do this because the question that you are saying was proved is "what does the sequence of symbols a ÷ b × c mean". The meaning of sequences of symbols is not a fundamental aspect of the universe, is it. They could have different meanings to different people, in different places, or at different times, couldn't they?