Distributing operations over =

x * (a=b) => x*a = x*b
x + (a=b) => x+a = x+b
(a=b)^2 => a^2 = b^2

so why do we have

x . (a=b) => x . (a=b) (??)
rather than x . a = x . b

(a=b)^^2 => (a=b)^^2 (??)

etc.

I do understand why we have

f(a=b) => f(a=b) rather than f(a)=f(b)

... because in general f may be an operation on
equations. Some operations nonetheless do distribute
over "=", notably

diff, taylor, limit,
cabs, rectform, realpart, imagpart, etc.
integrate is particularly clever, adding an
integration constant

though mathematical functions such as sin, log, etc. do
not.