<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> holds  <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Olive" title="b2">b</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span></span><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 <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span></span><br/>



<b>reconsider </b><font color="Maroon" title="c4">A</font> = <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of 1,<font color="Maroon" title="c1">K</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">B</font> = <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of 1,<font color="Maroon" title="c1">K</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">D</font> = <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of 1,<font color="Maroon" title="c1">K</font> ;<br/>
<a NAME="E4:28"/><i><font color="Green" title="E13">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c4">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c5">B</font> as   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font> ;<br/>
<a NAME="E6:28"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E15">A3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,1</span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:28_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E4:28"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">A</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T5" title="MATRIX_2:th.5">MATRIX_2:5</a></i>
;<br/>
<b>hence </b><a NAME="E4:28_1"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E1:28_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:28"/><i><font color="Green" title="E15">A4</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<i><font color="Green" title="E16">A5</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">B</font>,1</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">B</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:28_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E4:28"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:28_2"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">B</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T5" title="MATRIX_2:th.5">MATRIX_2:5</a></i>
;<br/>
<b>hence </b><a NAME="E5:28_2"/>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E1:28_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:28"/><i><font color="Green" title="E16">A6</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="matrix_3.html#E4:28"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="matrix_3.html#D4" target="_self" title="MATRIX_3:def.4">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c7">C</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of 1,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></i>;<br/>
<a NAME="E10:28"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">C</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:28"/><b>then </b><i><font color="Green" title="E17">A7</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E8:28"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">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="c8">C</font> implies <font color="Maroon" title="c8">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> )</span><br/><b>assume </b><a NAME="E1:28_3"/><i><font color="Green" title="E18">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">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="c8">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><br/><a NAME="E2:28_3"/><b>then </b>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 )
 <b>by </b><i><a class="txt" href="matrix_3.html#E11:28"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:28_3"/><b>then </b><i><font color="Green" title="E19">A9</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon" title="c8">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> = 
<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
<b>by </b><i><a class="txt" href="matrix_3.html#E4:28"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="matrix_3.html#E6:28"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="matrix_3.html#E7:28"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="matrix_3.html#E1:28_3"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="matrix_3.html#D4" target="_self" title="MATRIX_3:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="ref" href="fvsum_1.html#T113" title="FVSUM_1:th.113">FVSUM_1:113</a></i>
<br/>.= 
<span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E3:28_3"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="matrix_2.html#T5" title="MATRIX_2:th.5">MATRIX_2:5</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:28"/>
<span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
