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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green">E16</font></i>: 
(  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:22"/>
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:22_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<a NAME="E3:22_1"/>
<span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:22"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E4:22_1"/><b>then </b>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E2:22_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1:22"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T12">FINSEQ_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:22"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:22"/>
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 is   <a href="newton.html#NM1">Prime</a>
 <b>by </b><i><a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>


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