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In each part of maths( Algebra, Geometry etc) we have set some axioms that we accept as true without proving them and, using these axioms we go on proving statements, some of which are important and are called theorems. When we prove a theorem , we usually refer to a general state and speak generally, for example, take the Bolzano theorem for the continuous functions:" if f a continuous real function defined in [a,b] and

f(a)*f(b)<0 then there's at least one real number x with a<x<b so that f(x)=0

This means that any function that is continuous in [a,b] where a,b two numbers and f(a)f(b)<0 then at least one number x exists with a<x<b so that f(x)=0, right? How are we sure that any function that has the properties mentioned in the theorem has one point in which its value is zero? Of course in order to prove the theorem we don't use a specific function. This is what makes us sure that it applies in every function with these properties? The fact that we prove that a function f that is continuous in a set [a,b] with a ,b real numbers that we don't know and that f(a)f(b)<0 without knowing anything else about the values of the function makes us sure that what we do in order to prove this statement can be applied in any specific function with these properties, so we generalise the statement and say that we can use it in any real function? That's why theorems are general? This is the concept behind any theorem? And one more thing: if there were some functions in which this theorem wan't true, then this should definitely come up while trying to prove it?I mean some restrictions should be taken in order to complete the proof ,and these restrictions whould show us which are the functions that this theorem doen't apply to?

I referred to this theorem as an exampe but all these thoughts are about any theorem-statement that we prove in mathematics

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# Proving theorems

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