Definition
Mathematical proof
A deduction from stated assumptions and axioms using specified inference rules.
A mathematical proof is a deduction of a statement from specified hypotheses and axioms using accepted inference rules. In a formal proof, each step has an explicitly checkable justification. An ordinary written proof abbreviates such reasoning and may invoke previously established results.
Dependency of a conclusion
An assumption remains a hypothesis of the conclusion unless it is discharged by an inference rule. For example, deriving under an additional assumption proves after discharging . It does not prove without that condition.
Parameters
To prove a universal statement, an introduced parameter must be arbitrary under the permitted assumptions. To prove existence, a constructed object must be shown to satisfy every asserted property. Numerical examples and plausibility arguments can guide a proof but do not replace these steps.