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<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">d</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green">E41</font></i>: 
( not <font color="Maroon">m</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> &amp; <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> )
 ;<br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font>
 ;<br/>

<b>consider </b><font color="Maroon">z</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:39"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E5:39"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:39"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/>
<b>hence </b><a NAME="E7:39"/>
<font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>

<b>assume </b><a NAME="E8:39"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:39"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font>
<b>by </b><i>, , <a class="ref" href="nat_1.html#T49">NAT_1:49</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="wsierp_1.html#T2">WSIERP_1:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z</font></span>)</span>

;<br/>
<a NAME="E12:39"/><b>then </b>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:39"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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