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Example

Proposition: Between any two rational numbers, there is another rational number:

forall x in Q, y in Q: x < y => exists z in Q: x < z < y.

Proof: Take arbitrary x in Q and y in Q with x < y. Then x < (x+y)/2 < y because ...

Proof: Take arbitrary x in Q and y in Q with x < y. Let a := (x+y)/2. Then x < a < y because ...

Proof: Take arbitrary x in Q and y in Q with x < y. Let z := (x+y)/2. Then x < z < y because ...