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====C<sub>1</sub>. Double negation theorem====
====C<sub>1</sub>. Double negation theorem====
The first theorem goes under the names of ''Consequence 1'' (C<sub>1</sub>), the ''double negation theorem'' (DNT), or ''Reflection''.
The first theorem goes under the names of ''Consequence 1'' (C<sub>1</sub>), the ''double negation theorem'' (DNT), or ''Reflection''.
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====C<sub>2</sub>. Generation theorem====
====C<sub>2</sub>. Generation theorem====
One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the name of ''Consequence 2'' (C<sub>2</sub>), or ''Generation''.
One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the name of ''Consequence 2'' (C<sub>2</sub>), or ''Generation''.
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Here is a proof of the Generation Theorem.
Here is a proof of the Generation Theorem.
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| [[Image:Logical_Graph_Figure_28.jpg|500px]] || (28)
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====C<sub>3</sub>. Dominant form theorem====
====C<sub>3</sub>. Dominant form theorem====
The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence 3'' (C<sub>3</sub>), or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs.
The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence 3'' (C<sub>3</sub>), or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs.
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