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

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">M</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>;<br/>



<b>thus </b><a NAME="E1:112"/>
(  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies (  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> ) &amp; ( for <font color="Olive" title="b1">i0</font>, <font color="Olive" title="b2">j0</font>, <font color="Olive" title="b3">n0</font>, <font color="Olive" title="b4">m0</font> being non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j0</font> &amp; <font color="Olive" title="b3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">m0</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Olive" title="b1">i0</font>,<font color="Olive" title="b2">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Olive" title="b3">n0</font>,<font color="Olive" title="b4">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font> holds <br/> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Olive" title="b1">i0</font>,<font color="Olive" title="b2">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Olive" title="b3">n0</font>,<font color="Olive" title="b4">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:112_1"/><i><font color="Green" title="E97">A1</font></i>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>hence </b><a NAME="E2:112_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )
 <b>by </b><i><a class="ref" href="matrix13.html#T94" target="_self" title="MATRIX13:th.94">Th94</a></i>;<br/>

<b>let </b><font color="Maroon" title="c3">i0</font>, <font color="Maroon" title="c4">j0</font>, <font color="Maroon" title="c5">n0</font>, <font color="Maroon" title="c6">m0</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E3:112_1"/><i><font color="Green" title="E98">A2</font></i>: 
( <font color="Maroon" title="c3">i0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j0</font> &amp; <font color="Maroon" title="c5">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">m0</font> )
 <b>and </b><br/><a NAME="E4:112_1"/><i><font color="Green" title="E99">A3</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c4">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c5">n0</font>,<font color="Maroon" title="c6">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font>
 ;<br/>

<b>assume </b><a NAME="E5:112_1"/><i><font color="Green" title="E100">A4</font></i>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c4">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c5">n0</font>,<font color="Maroon" title="c6">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 ;<br/>

<a NAME="E6:112_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c4">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c5">n0</font>,<font color="Maroon" title="c6">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<b>by </b><i><a class="txt" href="matrix13.html#E3:112_1"><i><font color="Green" title="E98">A2</font></i></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>;<br/>
<b>hence </b><a NAME="E7:112_1"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix13.html#E1:112_1"><i><font color="Green" title="E97">A1</font></i></a>, <a class="txt" href="matrix13.html#E4:112_1"><i><font color="Green" title="E99">A3</font></i></a>, <a class="txt" href="matrix13.html#E5:112_1"><i><font color="Green" title="E100">A4</font></i></a>, <a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:112"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )
 ;<br/>

<a NAME="E3:112"/><b>then </b><i><font color="Green" title="E97">A5</font></i>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="matrix13.html#T94" target="_self" title="MATRIX13:th.94">Th94</a></i>;<br/>
<b>assume </b><a NAME="E4:112"/><i><font color="Green" title="E98">A6</font></i>: 
for <font color="Olive" title="b1">i0</font>, <font color="Olive" title="b2">j0</font>, <font color="Olive" title="b3">n0</font>, <font color="Olive" title="b4">m0</font> being non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j0</font> &amp; <font color="Olive" title="b3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">m0</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Olive" title="b1">i0</font>,<font color="Olive" title="b2">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Olive" title="b3">n0</font>,<font color="Olive" title="b4">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font> holds <br/> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Olive" title="b1">i0</font>,<font color="Olive" title="b2">j0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Olive" title="b3">n0</font>,<font color="Olive" title="b4">m0</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 ;<br/>

<b>assume </b><a NAME="E5:112"/><i><font color="Green" title="E99">A7</font></i>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:112"/><i><font color="Green" title="E100">A8</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P</font> <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="c4">Q</font> )
 <b>and </b><br/><a NAME="E8:112"/><i><font color="Green" title="E101">A9</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c2">M</font> &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )
 <b>by </b><i><a class="ref" href="matrix13.html#D4" target="_self" title="MATRIX13:def.4">Def4</a></i>;<br/>
<a NAME="E9:112"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="matrix13.html#E3:112"><i><font color="Green" title="E97">A5</font></i></a>, <a class="txt" href="matrix13.html#E5:112"><i><font color="Green" title="E99">A7</font></i></a>, <a class="txt" href="matrix13.html#E8:112"><i><font color="Green" title="E101">A9</font></i></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/>
<a NAME="E10:112"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">P1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">P</font><b> such that </b><br/><a NAME="E12:112"/><i><font color="Green" title="E102">A10</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="pencil_4.html#T2" title="PENCIL_4:th.2">PENCIL_4:2</a></i>;<br/>
<a NAME="E13:112"/>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D1" title="PSCOMP_1:def.1">PSCOMP_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">P1</font> = <font color="Maroon" title="c5">P1</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="pscomp_1.html#D1" title="PSCOMP_1:def.1">PSCOMP_1:def 1</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:112"/><i><font color="Green" title="E103">A11</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c6">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E12:112"><i><font color="Green" title="E102">A10</font></i></a>, <a class="ref" href="card_2.html#T79" title="CARD_2:th.79">CARD_2:79</a></i>;<br/>
<a NAME="E17:112"/>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> &amp; not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E16:112"><i><font color="Green" title="E103">A11</font></i></a>, <a class="ref" href="pscomp_1.html#D1" title="PSCOMP_1:def.1">PSCOMP_1:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c8">j</font> as  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">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>  ;<br/>
<b>consider </b><font color="Maroon" title="c11">Q1</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E20:112"/><i><font color="Green" title="E104">A12</font></i>: 
( <font color="Maroon" title="c11">Q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">P1</font> <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="c11">Q1</font> &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c6">P1</font>,<font color="Maroon" title="c11">Q1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E7:112"><i><font color="Green" title="E100">A8</font></i></a>, <a class="txt" href="matrix13.html#E8:112"><i><font color="Green" title="E101">A9</font></i></a>, <a class="ref" href="matrix13.html#T65" target="_self" title="MATRIX13:th.65">Th65</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E22:112"/><i><font color="Green" title="E105">A13</font></i>: 
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c11">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E12:112"><i><font color="Green" title="E102">A10</font></i></a>, <a class="txt" href="matrix13.html#E20:112"><i><font color="Green" title="E104">A12</font></i></a>, <a class="ref" href="card_2.html#T79" title="CARD_2:th.79">CARD_2:79</a></i>;<br/>
<a NAME="E23:112"/>
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q1</font> &amp; <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q1</font> &amp; not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q1</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E22:112"><i><font color="Green" title="E105">A13</font></i></a>, <a class="ref" href="pscomp_1.html#D1" title="PSCOMP_1:def.1">PSCOMP_1:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">n</font> = <font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c15">m</font> = <font color="Maroon" title="c13">m</font> as  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">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>  ;<br/>
<a NAME="E25:112"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">P1</font>,<font color="Maroon" title="c11">Q1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="matrix13.html#E20:112"><i><font color="Green" title="E104">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E26:112"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p2"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c15">m</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E7:112"><i><font color="Green" title="E100">A8</font></i></a>, <a class="txt" href="matrix13.html#E16:112"><i><font color="Green" title="E103">A11</font></i></a>, <a class="txt" href="matrix13.html#E22:112"><i><font color="Green" title="E105">A13</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E27:112"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix13.html#E4:112"><i><font color="Green" title="E98">A6</font></i></a>, <a class="txt" href="matrix13.html#E16:112"><i><font color="Green" title="E103">A11</font></i></a>, <a class="txt" href="matrix13.html#E20:112"><i><font color="Green" title="E104">A12</font></i></a>, <a class="txt" href="matrix13.html#E22:112"><i><font color="Green" title="E105">A13</font></i></a></i>;<br/>


</div>
