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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>  st <font color="Maroon" title="c1">p</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">r1</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  st <br/>( <font color="Olive" title="b2">p1</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b2">p1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">x</font> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b2">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> implies  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">r1</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  st <br/>( <font color="Olive" title="b1">p1</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b1">p1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b1">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c1">p</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">and </span><br/><a NAME="E3:23"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">r1</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  st <br/>( <font color="Olive" title="b1">p1</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b1">p1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b1">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">nx</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E5:23"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">nx</font>
 <span class="kw">and </span><br/><a NAME="E6:23"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="int_7.html#D1" target="_self" title="INT_7:def.1">Def1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">mx</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:23"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c3">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">mx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/>
<a NAME="E9:23"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c4">mx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/>
<a NAME="E10:23"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E11:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#D6" title="NEWTON:def.6">NEWTON:def 6</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">r10</font> = <span class="p1">(<span class="default"><a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E12:23"/>
<font color="Maroon" title="c2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E13:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A10</font></span>: 
 ex <font color="Olive" title="b1">r1</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  st <br/>( <font color="Olive" title="b1">r1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_7.html#E21"><span class="lab"><font color="Green" title="E19">Lm8</font></span></a></span>;<br/>
<a NAME="E14:23"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">r</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> <span class="kw"> such that </span><br/><a NAME="E16:23"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E17:23"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E18:23"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">and </span><br/><a NAME="E19:23"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E20:23"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c7">r</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E21:23"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c7">r</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="int_7.html#E7"><span class="lab"><font color="Green" title="E7">Lm3</font></span></a></span>;<br/>
<a NAME="E22:23"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c7">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:23"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E40">A19</font></span>: <font color="Maroon" title="c7">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:23"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="txt" href="#E22:23"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E41">A20</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">mx</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">mx</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">rxx</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">r20</font> = <span class="p1">(<span class="default"><a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">rxx</font>;<br/>
<a NAME="E26:23"/>
<font color="Maroon" title="c8">rxx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_3"/><span class="lab"><font color="Green" title="E42">A21</font></span>: 
<font color="Maroon" title="c8">rxx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<font color="Maroon" title="c7">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c8">rxx</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_3"><span class="lab"><font color="Green" title="E42">A21</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:23_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E23:23"><span class="lab"><font color="Green" title="E40">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">r2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> <span class="kw"> such that </span><br/><a NAME="E28:23"/><span class="lab"><font color="Green" title="E42">A22</font></span>: 
<font color="Maroon" title="c10">r2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">rxx</font>
 <span class="kw">and </span><br/><a NAME="E29:23"/><span class="lab"><font color="Green" title="E43">A23</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E30:23"/><span class="lab"><font color="Green" title="E44">A24</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rxx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="int_7.html#E21"><span class="lab"><font color="Green" title="E19">Lm8</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">p1</font> = <font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font>;<br/>
<a NAME="E31:23"/><span class="lab"><font color="Green" title="E45">A25</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:23"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="int_7.html#T1" target="_self" title="INT_7:th.1">Th1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E46">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> )</span><br/><a NAME="E1:23_4"/><span class="lab"><font color="Green" title="E47">A27</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>;<br/><span class="kw">assume </span><a NAME="E2:23_4"/><span class="lab"><font color="Green" title="E48">A28</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span><br/><a NAME="E3:23_4"/><span class="kw">then </span>
not <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:23_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A29</font></span>: 
<font color="Maroon" title="c10">r2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><a NAME="E5:23_4"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_4"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E6:23_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c10">r2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T38" title="PRE_POLY:th.38">PRE_POLY:38</a></span>;<br/><a NAME="E7:23_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A30</font></span>: 
( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font> or <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c10">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c6">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E2:23_4"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:23"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E2:23_4"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="txt" href="#E7:23_4"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:23_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_4"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="txt" href="#E4:23_4"><span class="lab"><font color="Green" title="E49">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> = 
 <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
;<br/>
<a NAME="E34:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:23"><span class="lab"><font color="Green" title="E46">A26</font></span></a>, <a class="ref" href="funct_1.html#T166" title="FUNCT_1:th.166">FUNCT_1:166</a></span>;<br/>
<a NAME="E35:23"/><span class="lab"><font color="Green" title="E48">A32</font></span>: 
<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:23"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">z</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> implies  ex <font color="Olive" title="b1">n</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/><a NAME="E1:23_6"/><span class="lab"><font color="Green" title="E49">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>;<br/><span class="kw">assume </span><a NAME="E2:23_6"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><a NAME="E3:23_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A34</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c10">r2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T38" title="PRE_POLY:th.38">PRE_POLY:38</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:23_6_1"/>
( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font> or <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c10">r2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_6"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_6_1_1"/><span class="lab"><font color="Green" title="E51">A35</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><div class="add"><a NAME="E2:23_6_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:23_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A36</font></span>: 
<font color="Maroon" title="c10">r2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><a NAME="E4:23_6_1_1"/><span class="lab"><font color="Green" title="E53">A37</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><font color="Maroon" title="c6">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_6_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:23"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E1:23_6_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:23_6_1_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:23_6"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="txt" href="#E1:23_6_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E4:23_6_1_1"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="txt" href="#E3:23_6_1_1"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="int_7.html#D1" target="_self" title="INT_7:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_6_1_2"/><span class="lab"><font color="Green" title="E51">A38</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c10">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">mx</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">mx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">mx</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">mx</font> = <font color="Maroon" title="c4">mx</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">mx</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">mx</font> )</span><br/><span class="kw">thus </span><a NAME="E2:23_6_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">mx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">mx</font></span><br/><a NAME="E3:23_6_1_2"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E1:23_6_1_2"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:23_6_1_2"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
not <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:23"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E35:23"><span class="lab"><font color="Green" title="E48">A32</font></span></a>, <a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E1:23_6_1_2"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_6_1_2"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c12">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">mx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:23"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E28:23"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E1:23_6_1_2"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E3:23_6_1_2"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:23_6_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">mx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E37:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A41</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#D1" target="_self" title="INT_7:def.1">Def1</a></span>;<br/>
<a NAME="E38:23"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E39:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A42</font></span>: 
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:23"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c8">rxx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:23"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></span>
;<br/>
<a NAME="E41:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A43</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:23"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<a NAME="E42:23"/>
<font color="Maroon" title="c7">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rxx</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>;<br/>
<a NAME="E43:23"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">rr1</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  st <br/>( <font color="Olive" title="b1">rr1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">r</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">rr1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b1">rr1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rxx</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E23:23"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E31:23"><span class="lab"><font color="Green" title="E45">A25</font></span></a>, <a class="txt" href="int_7.html#E21"><span class="lab"><font color="Green" title="E19">Lm8</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E52">A44</font></span>:  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> = 
<span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:23"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E21:23"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="txt" href="#E30:23"><span class="lab"><font color="Green" title="E44">A24</font></span></a>, <a class="ref" href="nat_3.html#T19" title="NAT_3:th.19">NAT_3:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c10">r2</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:23"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E18:23"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="nat_3.html#T19" title="NAT_3:th.19">NAT_3:19</a></span>
;<br/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c10">r2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E29:23"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="pre_poly.html#T38" title="PRE_POLY:th.38">PRE_POLY:38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
<br/>.= 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:23"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E46:23"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">r1</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  st <br/>( <font color="Olive" title="b1">p1</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b1">p1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b1">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E41:23"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="txt" href="#E44:23"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="txt" href="#E37:23"><span class="lab"><font color="Green" title="E49">A41</font></span></a>, <a class="txt" href="#E34:23"><span class="lab"><font color="Green" title="E47">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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