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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="pepin.html#K1" target="_self">^2</a>  = 
<font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="nat_1.html#T69">NAT_1:69</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span>

<br/>.= 
<font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E3:40"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="pepin.html#K1" target="_self">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font>
 ;<br/>


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