Line 1,094:
Line 1,094:
It remains to show that the right hand side of the equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}</math> is logically equivalent to the DNF just obtained. The needed chain of equations is as follows:
It remains to show that the right hand side of the equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}</math> is logically equivalent to the DNF just obtained. The needed chain of equations is as follows:
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{| align="center" cellpadding="8" style="text-align:center; width:90%"
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{| align="center" cellpadding="8"
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<pre>
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
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o-----------------------------------------------------------o
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|-
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| Equation E_1. Proof 2, 2nd Half. |
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-0.jpg|500px]]
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o-----------------------------------------------------------o
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|-
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| |
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-1.jpg|500px]]
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| q r |
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|-
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| o |
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| [[Image:Equational Inference Bar -- Cast P.jpg|500px]]
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| | |
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|-
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| p o |
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-2.jpg|500px]]
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| | |
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|-
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| @ |
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| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
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| |
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|-
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o=============================< CAST "p" >==================o
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-3.jpg|500px]]
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| |
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|-
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| q r q r |
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| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
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| o o o |
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|-
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| | \| |
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-4.jpg|500px]]
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| o o |
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|-
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| | | |
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| [[Image:Equational Inference Bar -- Cast Q.jpg|500px]]
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| p o-------o---o p |
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|-
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| \ / |
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-5.jpg|500px]]
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| \ / |
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|-
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| \ / |
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| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
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| @ |
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|-
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| |
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-6.jpg|500px]]
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o=============================< Domination >================o
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|-
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| |
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| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
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| q r |
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|-
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| o o |
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-7.jpg|500px]]
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| | \ |
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|-
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| o o |
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| [[Image:Equational Inference Bar -- Cast R.jpg|500px]]
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| | | |
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|-
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| p o-------o---o p |
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-8.jpg|500px]]
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| \ / |
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|-
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| \ / |
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| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
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| \ / |
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|-
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| @ |
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-9.jpg|500px]]
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| |
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|-
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o=============================< Cancellation >==============o
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| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
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| |
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|}
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| q r |
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| o |
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| | |
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| o |
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| | |
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| p o-------o---o p |
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| \ / |
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| \ / |
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| \ / |
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| @ |
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| |
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o=============================< CAST "q" >==================o
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| |
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| o |
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| | |
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| o r o r |
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| | | |
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| o o |
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| | | |
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| q o-------o---o q |
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| \ / |
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| \ / |
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| \ / |
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| p o-------o---o p |
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| \ / |
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| \ / |
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| \ / |
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| @ |
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| |
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o=============================< Domination >================o
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| |
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| o |
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| | |
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| o r o |
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| | | |
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| o o |
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| | | |
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| q o-------o---o q |
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| \ / |
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| \ / |
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| \ / |
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| p o-------o---o p |
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| \ / |
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| \ / |
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| \ / |
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| @ |
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| |
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o=============================< Cancellation >==============o
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| |
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| o r |
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| | |
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| o o |
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| | | |
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| q o-------o---o q |
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| \ / |
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| \ / |
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| \ / |
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| p o-------o---o p |
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| \ / |
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| \ / |
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| \ / |
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| @ |
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| |
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o=============================< CAST "r" >==================o
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| |
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| o |
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| | |
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| o o |
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| | | |
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| o o |
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| | | |
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| r o-------o---o r |
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| \ / |
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| \ / o |
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| \ / | |
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| q o-------o---o q |
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| \ / |
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| \ / |
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| \ / |
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| p o-------o---o p |
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| \ / |
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| \ / |
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| \ / |
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| @ |
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| |
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o=============================< Cancellation >==============o
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| |
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| o |
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| | |
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| r o-------o---o r |
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| \ / |
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| \ / o |
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| \ / | |
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| q o-------o---o q |
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| \ / |
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| \ / |
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| \ / |
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| p o-------o---o p |
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| \ / |
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| \ / |
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| \ / |
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| @ |
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| |
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o=============================< DNF >=======================o
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</pre>
| (33)
| (33)
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