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<div class="add">

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">o</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font>, <font color="Maroon">o'</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">o'</font> is <a href="binop_1.html#V1" target="_self">commutative</a>
 ;<br/>

<a NAME="E2:27"/>
( ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span> &amp; <font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span>,<font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> ) ) iff for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span>,<font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:27_1"/>
( ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span> &amp; <font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span>,<font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> ) ) implies for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span>,<font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E2:27_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span>,<font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<b>thus </b><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">a</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">c</font>,<font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">c</font>,<font color="Maroon">a</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i>, </i>
;<br/>

<b>thus </b><a NAME="E4:27_1"/>
<font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27"/>
( <font color="Maroon">o'</font> <a href="binop_1.html#R6" target="_self">is_distributive_wrt</a> <font color="Maroon">o</font> iff for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span>,<font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
