*If I turn out to have a wrong answer, please no hints or showing an valid proof. I want to do it on my own !

ad/bd-bc/bd=+-1/bd is neighbor fraction

Now, reduce the common numbers :

a/b-c/d=+-1/bd

We must now prove that the left hand side has irreductible fractions. Lets see what would happen if these fraction were reductible.

let a=z*y , b=z*l , c=p*m d=p*n

z*y/z*l-p*m/p*n equality to be determined +-1/(z*l)*(p*n)

Reduce:

*ln (y/l-m/n) equality to be determined (+-1/(z*l)*(p*n))*ln

yn-lm equality to be determined +-1/z*p

yn-lm not= +-1/z*p

We have considered the initial fractions to be reductible and have arrived at a false result. An integer cannot be equal to (+-1/z*p)

So, the initial fractions must be irreductible.

*Of course, I'm considering the variables to represents integers.