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congruent complements
One cannot have a theory (theorem) without proof. Theories are explanations (models with uses such as predicting on outcome of an experiment or event) for scientific laws, which only describe the phenomenon. Proof is anything that backs up a hypothesis. When a significant amount of proof is shown, the hypothesis becomes a theory due to it being accepting by the scientific community.
Proof
There are many ways of interpreting "contradiction" in mathematics. Some meanings are:Contradiction as in proof. You attempt to give the counter-proof of the theorem, but the counter-proof fails to work.Contradiction as in mathematical logic. If biconditional fails, we include the slash through the double arrows pointing left and right at opposite directions.Contradiction as in negation of the clause.
Usually. A coin in Proof condition is almost always more valuable than the same coin in Uncirculated condition, but exceptions do exist.