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<b>let </b><font color="Maroon">n</font> be  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="E1:20"/><i><font color="Green">E41</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1"/>
(  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> or  not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>suppose </b></a><a NAME="E1:20_1_1"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2"><b>suppose </b></a><a NAME="E1:20_1_2"/>
not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">S</font> =  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:20_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:20_1_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <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#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E2:20_1_2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> 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/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i>, , <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<a NAME="E4:20_1_2_1"/><i><font color="Green">E56</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:20_1_2_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:20_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>take </b>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font>
;<br/><a NAME="E4:20_1_2"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:20_1_2"/>
(  <a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font></span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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