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<b>let </b><font color="Maroon">A</font> be    <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>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font>;<br/>





<b>thus </b><a NAME="E1:25"/>
( <font color="Maroon">o'</font> <a href="binop_1.html#R6" target="_self">is_distributive_wrt</a> <font color="Maroon">o</font> 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 <br/> ( <font color="Maroon">o'</font> <a href="binop_1.html#K1" target="_self">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K1" 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#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K1" 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#K1" 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="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/>
( <font color="Maroon">o'</font> <a href="binop_1.html#R4" target="_self">is_left_distributive_wrt</a> <font color="Maroon">o</font> &amp; <font color="Maroon">o'</font> <a href="binop_1.html#R5" target="_self">is_right_distributive_wrt</a> <font color="Maroon">o</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D11" target="_self">BINOP_1:def 11</a><br/>

<b>hence </b><a NAME="E2:25_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#K1" target="_self">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K1" 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#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:25"/>
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#K1" target="_self">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K1" 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#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> )
 ;<br/>

<b>hence </b><a NAME="E3:25"/>
( ( 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#K1" target="_self">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <font color="Olive">a</font>,<font color="Olive">c</font></span>)</span> ) &amp; ( 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="binop_1.html#K1" 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#K1" target="_self">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D9" target="_self">BINOP_1:def 9</a>,<a class="ref" href="binop_1.html#D10" target="_self">BINOP_1:def 10</a>,<a class="ref" href="binop_1.html#D11" target="_self">BINOP_1:def 11</a><br/>


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