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"The product of x and of the sum of y and z if the sum of the products of x and y and of x and z":
x*(y+z) = x*y + x*z.
"Between every natural number and its double (the bounds included) there is at least one prime number":
forall x: x in N => exists y: x <= y /\ y <= 2x /\ y is prime.
Formal notation is a tool to write clear and concise statements.