<?xml version="1.0"?>
<div class="add">

<b>thus </b><a NAME="E1:42_1_1"/>
( <font color="Maroon">n</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> implies  ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:42_1_1"/>
( not <font color="Maroon">n</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> implies  ex <font color="Olive">n'</font> being non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Olive">n'</font></span>)</span></span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_1_1"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">n</font> is <a href="moebius1.html#V1" target="_self">square-containing</a>
 ;<br/>

<a NAME="E2:42_1_1_1"/>
<font color="Maroon">n</font> is  non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> is  not  non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> 
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a>;<br/>
<a NAME="E3:42_1_1_1_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</font> as  non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E4:42_1_1_1"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n'</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:42_1_1_1"/>
 ex <font color="Olive">n'</font> being non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Olive">n'</font></span>)</span></span>)</span></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:42_1_1"/>
( not <font color="Maroon">n</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> implies  ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ex <font color="Olive">n'</font> being non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Olive">n'</font></span>)</span></span>)</span></span>)</span> ) )
 ;<br/>


</div>
