I’m not able to demonstrate, with mathematical steps, that if the test for a disease has no false negatives, written as P(M,-T) = 0, then P(T|M) = 1, meaning that the probability of the test being positive given that the patient has the disease is 1.

I can understand it locally, but I would like to demonstrate it analytically.

Here’s my attempt at proving that P(T|M) = 1:

Starting from the given information:
[1] P(M, -T) = 0: The test has no false negatives, so the probability of a negative result given that the patient has the disease is zero.

Let’s define the probabilities as follows:
P(T|M): Probability of a positive result given that the patient has the disease
P(M): Probability that the patient has the disease
P(M|T): Probability that the patient has the disease given a positive result
P(T): Probability of a positive result (regardless of the disease)

Relations between probabilities:
[2] Bayes’ Theorem: P(M|T) * P(T) = P(T|M) * P(M)
[3] Total Probability: P(T) = P(M,T) + P(M,-T)

For [1] and [3], substituting P(M,-T) = 0 into the definition of P(T):
[4] P(T) = P(M,T)

Using [2] and [4], substituting P(T) into the relation between probabilities:
[5] P(M|T) * P(M,T) = P(T|M) * P(M)

Dividing both sides by P(M,T):
[6] P(M|T) = P(T|M) * P(M) / P(M,T)

[7] P(M|T) / P(M) = P(T|M) / P(M,T)

…