B Question about properties of sums and products

[MAGNIBORO]

Hi and sorry for bad english.I want to know if these properties are truehttps://gyazo.com/88c471f4bb9989b67a390c372f2c72fe

and

https://gyazo.com/85f4110664db6831576012debaf3a778 I did not find these properties in any place.
so I guess it will be obvious or are incorrect,
if incorrect I would like to see counterexamples.
thanks

 

[micromass]

Can you write these properties using an "if ... then" structure and not use the word "so"?

Also, can you write basic definitions of $a$, $b$, $f$ and so on? Are we working in the real numbers? Or a general vector space? Or?

 

[MAGNIBORO]

Can you write these properties using an "if ... then" structure and not use the word "so"?

Also, can you write basic definitions of $a$, $b$, $f$ and so on? Are we working in the real numbers? Or a general vector space? Or?

a and b are any whole numbers (or infinity) , f(i), g(i),t(i),b(i) are any functions of i

 

[whit3r0se-]

For the first case, it isn't always true.
The Sum of x between 1 and 0 is 1, and the sum of x^2 between 1 and 0 is also one. We can generalise this to x^n where n is real and not equal to 0, this would produce the same answer between [0,1]