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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/>( ( <font color="Olive">p</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> ) &amp; ( not <font color="Olive">p</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) );<br/>
<a NAME="E1:95_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:95_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 <b>and </b><br/><a NAME="E2:95_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>]
 <b>and </b><br/><a NAME="E3:95_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y2</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="txt" href="nat_3.html#E2:95_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/>
<a NAME="E5:95_1_1_1"/>
( ( <font color="Maroon">p</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> implies <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> ) &amp; ( not <font color="Maroon">p</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> implies <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0 ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:95_1_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:95_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:95_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="txt" href="nat_3.html#E1:95_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_1_2_1"/>
( <font color="Maroon">i</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> or  not <font color="Maroon">i</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:95_1_1_2_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span>
;<br/><b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>assume </b><a NAME="E2:95_1_1_2_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><b>hence </b><a NAME="E3:95_1_1_2_1_1"/>
( ( <font color="Maroon">p</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> implies <font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> ) &amp; ( not <font color="Maroon">p</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> implies <font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="txt" href="nat_3.html#E1:95_1_1_2_1_1"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:95_1_1_2_1_2"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">i</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><b>take </b>
0
;<br/><b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>assume </b><a NAME="E2:95_1_1_2_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><b>hence </b><a NAME="E3:95_1_1_2_1_2"/>
( ( <font color="Maroon">p</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> implies 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> ) &amp; ( not <font color="Maroon">p</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> implies 0 <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:95_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="newton.html#K13">SetPrimes</a> 
 <b>and </b><br/><a NAME="E5:95_1_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, <a class="txt" href="nat_3.html#E2:95_1_1"><i><font color="Green">E50</font></i></a>);<br/></i>
<a NAME="E6:95_1_1"/><i><font color="Green">E103</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K13">SetPrimes</a> 
 
<b>by </b><i>, <a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/>
<div><i><font color="Green">E104</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:95_1_1_3_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:95_1_1_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/><b>assume </b><a NAME="E4:95_1_1_3_1"/>
not <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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:95_1_1_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="nat_3.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E6:95_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_3.html#T4" target="_self">Th4</a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:95_1_1_3"/><i><font color="Green">E106</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E3:95_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/><a NAME="E5:95_1_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E6:95_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="newton.html#K13">SetPrimes</a>  <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E9:95_1_1"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font> <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#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

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

<b>assume </b><a NAME="E10:95_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">p</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#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E11:95_1_1"/><b>then </b>
<font color="Maroon">p</font> is   <a href="newton.html#NM1">Prime</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:95_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T4" target="_self">Th4</a></i>;<br/>


</div>
