direct proof of the theorem

a$.$ The difference between two distinct real numbers is not equal to zero.

b$.$ The product of two positive real numbers is a positive number

c$.$ The average of two real numbers is greater than or equal to at least one of the numbers.

Let $$ and $$ be real numbers such that $>0$ and $>0.$

Let $$ and $$ be real numbers.

Let $$ and $$ be real numbers such that $!=.$