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

<span class="kw">reconsider </span><font color="Maroon" title="c1">x1</font> = 1, <font color="Maroon" title="c2">x2</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">r</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">p</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font> &amp; not <font color="Olive" title="b1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Olive" title="b1">p</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><br/><span class="kw">let </span><font color="Maroon" title="c6">p</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font> &amp; not <font color="Maroon" title="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">p</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><br/>



<span class="kw">assume </span><a NAME="E2:42"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font>
 ; <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="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">p</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><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">n9</font> = <font color="Maroon" title="c3">n</font> as   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">k2</font> = <font color="Maroon" title="c4">m</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">reconsider </span><font color="Maroon" title="c9">p9</font> = <font color="Maroon" title="c6">p</font> as   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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">reconsider </span><font color="Maroon" title="c10">k1</font> = <font color="Maroon" title="c9">p9</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <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="c11">S</font> =  <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">CS</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">T</font> =  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c8">k2</font>;<br/>
<a NAME="E7:42"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c8">k2</font></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="Maroon" title="c8">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></span>;<br/>
<a NAME="E8:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c8">k2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:42"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
not <font color="Maroon" title="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font>
 ; <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="c3">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">p</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><br/>

<a NAME="E10:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c8">k2</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="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">T</font> =  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c8">k2</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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/>
<span class="kw">set </span><font color="Maroon" title="c15">X</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><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="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<font color="Maroon" title="c14">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">X9</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><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="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<font color="Maroon" title="c14">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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/>
<span class="kw">set </span><font color="Maroon" title="c17">E</font> =  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c10">k1</font>,<font color="Maroon" title="c16">X9</font>;<br/>
<a NAME="E13:42"/>
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K3" title="GR_CY_1:func.3">addint</a> <font color="Maroon" title="c10">k1</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_1.html#D6" title="GR_CY_1:def.6">GR_CY_1:def 6</a></span>;<br/>
<a NAME="E14:42"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></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="Maroon" title="c10">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></span>;<br/>
<a NAME="E15:42"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/>
<a NAME="E16:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A5</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><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="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<font color="Maroon" title="c14">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="card_2.html#T65" title="CARD_2:th.65">CARD_2:65</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_1"/>
<font color="Maroon" title="c6">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E3:42_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">n9</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">n9</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n9</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">n9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/><a NAME="E4:42_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n9</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">n9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E5:42_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n9</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">n9</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></span>;<br/><a NAME="E6:42_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E7:42_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><span class="kw">hence </span><a NAME="E8:42_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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 NAME="E18:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A6</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">k1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<font color="Maroon" title="c14">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="nat_1.html#T41" title="NAT_1:th.41">NAT_1:41</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">E9</font> =  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c10">k1</font>,<font color="Maroon" title="c16">X9</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#T2" target="_self" title="GROUP_10:th.2">Th2</a></span>;<br/>
<a NAME="E20:42"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<font color="Maroon" title="c14">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c10">k1</font>,<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></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="Maroon" title="c18">E9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#T3" target="_self" title="GROUP_10:th.3">Th3</a></span>;<br/>
<a NAME="E21:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A7</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">E9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c10">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="card_fin.html#T18" title="CARD_FIN:th.18">CARD_FIN:18</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">LO</font> =  <a href="group_10.html#K7" title="GROUP_10:func.7">the_left_operation_of</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<font color="Maroon" title="c14">T</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">LO9</font> =  <a href="group_10.html#K5" title="GROUP_10:func.5">the_extension_of_left_operation_of</a> <font color="Maroon" title="c10">k1</font>,<span class="p1">(<span class="default"><a href="group_10.html#K7" title="GROUP_10:func.7">the_left_operation_of</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<font color="Maroon" title="c14">T</font></span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E39">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_2"/>
<font color="Maroon" title="c4">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c9">p9</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/><a NAME="E2:42_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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 NAME="E23:42"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="group_10.html#E32"><span class="lab"><font color="Green" title="E28">Lm2</font></span></a></span>;<br/>
<span class="kw">then </span> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <span class="p2">(<span class="default"><a href="group_10.html#K5" title="GROUP_10:func.5">the_extension_of_left_operation_of</a> <font color="Maroon" title="c10">k1</font>,<span class="p3">(<span class="default"><a href="group_10.html#K7" title="GROUP_10:func.7">the_left_operation_of</a> <span class="p4">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span></span>)</span> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">T</font> : verum </span>}</span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:42"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="group_10.html#E34"><span class="lab"><font color="Green" title="E30">Lm4</font></span></a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">T</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="group_10.html#E33"><span class="lab"><font color="Green" title="E29">Lm3</font></span></a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">k2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></span>
;<br/>
<a NAME="E25:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A10</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <span class="p2">(<span class="default"><a href="group_10.html#K5" title="GROUP_10:func.5">the_extension_of_left_operation_of</a> <font color="Maroon" title="c10">k1</font>,<span class="p3">(<span class="default"><a href="group_10.html#K7" title="GROUP_10:func.7">the_left_operation_of</a> <span class="p4">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font></span>)</span>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/>
<a NAME="E26:42"/>
 <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c10">k1</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_p-group_of_prime</a> <font color="Maroon" title="c6">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:42"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:42"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">p</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="#E21:42"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E25:42"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E22:42"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="group_10.html#T11" target="_self" title="GROUP_10:th.11">Th11</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>
