Because it won't type check. It's like asking how you know some function in a program really gets an Int argument when you had lots of data structures you were passing around with all kinds of fields with various names and types.
If you construct value of type `forall n, exists m, m≥n and m prime`, then you know you have such an object. Think of a `theorem` (the Lean keyword) as a function that returns such objects. If you try to return the wrong type (e.g. you build a `forall n, exists m, m≤n and m prime`) in your "function" body, you get a compile error. If you use types incorrectly in the middle (passing values and intermediate proofs to theorems incorrectly), you get a compile error.
It's just like any programming language except the type system is rich enough that in addition to being able to have a value x: Int, (so maybe x is 5) you can have values like h: x^2≥0 (so imagine h is some object you build that proves x^2≥0).
If you construct value of type `forall n, exists m, m≥n and m prime`, then you know you have such an object. Think of a `theorem` (the Lean keyword) as a function that returns such objects. If you try to return the wrong type (e.g. you build a `forall n, exists m, m≤n and m prime`) in your "function" body, you get a compile error. If you use types incorrectly in the middle (passing values and intermediate proofs to theorems incorrectly), you get a compile error.
It's just like any programming language except the type system is rich enough that in addition to being able to have a value x: Int, (so maybe x is 5) you can have values like h: x^2≥0 (so imagine h is some object you build that proves x^2≥0).