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

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>





<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E39">A1</font></i>: 
(  <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="c4">n</font> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/>

<a NAME="E2:47"/><i><font color="Green" title="E40">A2</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<b>assume </b><a NAME="E3:47"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c3">b</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">L</font> =  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c5">L</font><b> such that </b><br/><a NAME="E6:47"/><i><font color="Green" title="E42">A4</font></i>: 
(  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">L</font><br/> for <font color="Olive" title="b2">xz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span><br/> for <font color="Olive" title="b3">xn</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">xz</font> &amp; <font color="Olive" title="b3">xn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">xz</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> holds <br/> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">xn</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="int_7.html#E1:47"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="int_7.html#E46"><i><font color="Green" title="E37">Lm14</font></i></a></i>;<br/>
<a NAME="E7:47"/><i><font color="Green" title="E43">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c7">x1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E2:47_1"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font>
 
;<br/>
<a NAME="E3:47_1"/><b>then </b>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</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="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="int_7.html#E2:47"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="int_7.html#D2" target="_self" title="INT_7:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">x3</font> = <font color="Maroon" title="c7">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">L</font> <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<i><font color="Green" title="E45">A7</font></i>: <font color="Maroon" title="c7">x1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="txt" href="int_7.html#E1:47_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="int_7.html#T21" target="_self" title="INT_7:th.21">Th21</a></i>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="ref" href="int_7.html#T21" target="_self" title="INT_7:th.21">Th21</a></i>
;<br/>
<a NAME="E6:47_1"/>
<font color="Maroon" title="c7">x1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_7.html#K1" title="INT_7:func.1">Segm0</a> <font color="Maroon" title="c1">p</font>
 
;<br/>
<a NAME="E7:47_1"/><b>then </b>
<font color="Maroon" title="c7">x1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</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="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="int_7.html#E2:47"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="int_7.html#D2" target="_self" title="INT_7:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">x2</font> = <font color="Maroon" title="c7">x1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">x3</font> = 
<font color="Maroon" title="c9">x2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_7.html#E6:47"><i><font color="Green" title="E42">A4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">L</font>
<b>by </b><i><a class="txt" href="int_7.html#E5:47_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
;<br/>
<a NAME="E10:47_1"/><b>then </b>
<font color="Maroon" title="c8">x3</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></i>;<br/>
<b>hence </b><a NAME="E11:47_1"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D9" title="POLYNOM5:def.9">POLYNOM5:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:47"/><i><font color="Green" title="E44">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_2"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/>

<a NAME="E2:47_2"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E3:47_2"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T49" title="GROUP_2:th.49">GROUP_2:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E5:47_2"/>
<font color="Maroon" title="c8">x1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:47_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E7:47_2"/><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T25" title="GR_CY_1:th.25">GR_CY_1:25</a></i>;<br/>
<font color="Maroon" title="c8">x1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> = 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_7.html#E7:47_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="group_1.html#T69" title="GROUP_1:th.69">GROUP_1:69</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c9">j</font>
<b>by </b><i><a class="ref" href="group_1.html#T68" title="GROUP_1:th.68">GROUP_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c9">j</font>
<b>by </b><i><a class="txt" href="int_7.html#E1:47"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="group_1.html#T82" title="GROUP_1:th.82">GROUP_1:82</a></i>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T61" title="GROUP_1:th.61">GROUP_1:61</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="int_7.html#T21" target="_self" title="INT_7:th.21">Th21</a></i>
;<br/>
<b>hence </b><a NAME="E9:47_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="int_7.html#E7:47"><i><font color="Green" title="E43">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c7">RF</font> =  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="int_7.html#E6:47"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="int_7.html#T27" target="_self" title="INT_7:th.27">Th27</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">m</font>, <font color="Maroon" title="c9">n0</font> being    <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> <b> such that </b><br/><a NAME="E11:47"/><i><font color="Green" title="E45">A11</font></i>: 
( <font color="Maroon" title="c9">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp; <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n0</font> )
 <b>by </b><i><a class="txt" href="int_7.html#E6:47"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="int_7.html#T27" target="_self" title="INT_7:th.27">Th27</a></i>;<br/>
<a NAME="E12:47"/><i><font color="Green" title="E46">A12</font></i>: 
 ex <font color="Olive" title="b1">D</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp;  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">D</font> )
 
;<br/>
<a NAME="E13:47"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:47"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="int_7.html#E7:47"><i><font color="Green" title="E43">A5</font></i></a></i>;<br/>
<a NAME="E14:47"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E15:47"/><b>then </b><i><font color="Green" title="E47">A13</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="int_7.html#E8:47"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">XX</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<i><font color="Green" title="E48">A14</font></i>:  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">XX</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="int_7.html#E3:47"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>
<br/>.= 
<font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="int_7.html#E1:47"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="int_7.html#E6:47"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="int_7.html#E11:47"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="int_7.html#E12:47"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></i>
;<br/>
<a NAME="E18:47"/><i><font color="Green" title="E49">A15</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E15:47"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E19:47"/><i><font color="Green" title="E50">A16</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E11:47"><i><font color="Green" title="E45">A11</font></i></a></i>;<br/>
<a NAME="E20:47"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E17:47"><i><font color="Green" title="E48">A14</font></i></a></i>;<br/>
<a NAME="E21:47"/><b>then </b>
<font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="txt" href="int_7.html#E18:47"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="int_7.html#E19:47"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="card_1.html#T72" title="CARD_1:th.72">CARD_1:72</a></i>;<br/>
<a NAME="E22:47"/><b>then </b>
<font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<b>hence </b><a NAME="E23:47"/>
contradiction
 <b>by </b><i><a class="txt" href="int_7.html#E11:47"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="nat_1.html#T16" title="NAT_1:th.16">NAT_1:16</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div>
