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<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">M</font> being  <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font> holds <br/> (  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff (  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">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="b2">i</font>,<font color="Olive" title="b3">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="Olive" title="b1">M</font> &amp; <font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">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="b2">i</font>, <font color="Olive" title="b3">j</font>, <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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="b2">i</font>,<font color="Olive" title="b3">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="Olive" title="b4">n</font>,<font color="Olive" title="b5">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="Olive" title="b1">M</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b4">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j</font>,<font color="Olive" title="b5">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b5">m</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j</font>,<font color="Olive" title="b4">n</font></span>)</span> ) ) )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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 iff (  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">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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">i</font>,<font color="Olive" title="b2">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="Olive" title="b3">n</font>,<font color="Olive" title="b4">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> holds <br/><span class="p1">(<span class="default"><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="b3">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="b4">m</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">n</font></span>)</span> ) ) )</span><br/>



<b>thus </b><a NAME="E1:113"/>
(  <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">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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">i</font>,<font color="Olive" title="b2">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="Olive" title="b3">n</font>,<font color="Olive" title="b4">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> holds <br/><span class="p1">(<span class="default"><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="b3">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="b4">m</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">n</font></span>)</span> ) ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  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">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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">i</font>,<font color="Olive" title="b2">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="Olive" title="b3">n</font>,<font color="Olive" title="b4">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> holds <br/><span class="p1">(<span class="default"><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="b3">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="b4">m</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">n</font></span>)</span> ) implies  <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 )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:113_1"/><i><font color="Green" title="E98">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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  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">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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">i</font>,<font color="Olive" title="b2">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="Olive" title="b3">n</font>,<font color="Olive" title="b4">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> holds <br/><span class="p1">(<span class="default"><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="b3">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="b4">m</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">n</font></span>)</span> ) )</span><br/>

<b>hence </b><a NAME="E2:113_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#T96" target="_self" title="MATRIX13:th.96">Th96</a></i>; <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">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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">i</font>,<font color="Olive" title="b2">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="Olive" title="b3">n</font>,<font color="Olive" title="b4">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> holds <br/><span class="p1">(<span class="default"><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="b3">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="b4">m</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">n</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font>, <font color="Maroon" title="c5">n</font>, <font color="Maroon" title="c6">p</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="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">i</font>,<font color="Maroon" title="c4">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="c5">n</font>,<font color="Maroon" title="c6">p</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> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E3:113_1"/><i><font color="Green" title="E99">A2</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">i</font>,<font color="Maroon" title="c4">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="c5">n</font>,<font color="Maroon" title="c6">p</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:113_1_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> or <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> or ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1"><b>suppose </b></a><a NAME="E1:113_1_1_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> or <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:113_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span>
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_2"><b>suppose </b></a><a NAME="E1:113_1_1_2"/><i><font color="Green" title="E100">A3</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span></span><br/><div class="add"><a NAME="E2:113_1_1_2"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</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> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <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="E3:113_1_1_2"/><b>then </b>
( <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">M</font></span>)</span> &amp; <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c6">p</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">M</font></span>)</span> &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c6">p</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> &amp; <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c6">p</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> &amp; not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c2">M</font></span>)</span> &amp; not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c2">M</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:113_1"><i><font color="Green" title="E99">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="zfmisc_1.html#T138" title="ZFMISC_1:th.138">ZFMISC_1:138</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">I</font> = <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c8">J</font> = <font color="Maroon" title="c4">j</font>, <font color="Maroon" title="c9">P</font> = <font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c10">N</font> = <font color="Maroon" title="c5">n</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>  ;<br/><a NAME="E5:113_1_1_2"/><i><font color="Green" title="E101">A4</font></i>: 
(  <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="c7">I</font>,<font color="Maroon" title="c8">J</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="c10">N</font>,<font color="Maroon" title="c9">P</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="#E1:113_1_1_2"><i><font color="Green" title="E100">A3</font></i></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>;<br/><a NAME="E6:113_1_1_2"/><b>then </b><i><font color="Green" title="E102">A5</font></i>: 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c8">J</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c10">N</font>,<font color="Maroon" title="c9">P</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c8">J</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>,<span class="p1"><a href="matrix13.html#K4" title="MATRIX13:func.4">{</a><span class="default"><font color="Maroon" title="c10">N</font>,<font color="Maroon" title="c9">P</font></span><a href="matrix13.html#K4" title="MATRIX13:func.4">}</a></span>
 <b>by </b><i><a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>;<br/><a NAME="E7:113_1_1_2"/><i><font color="Green" title="E103">A6</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="c7">I</font>,<font color="Maroon" title="c8">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="c10">N</font>,<font color="Maroon" title="c9">P</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>
 <b>by </b><i><a class="txt" href="#E1:113_1"><i><font color="Green" title="E98">A1</font></i></a>, <a class="txt" href="#E3:113_1"><i><font color="Green" title="E99">A2</font></i></a>, <a class="txt" href="#E1:113_1_1_2"><i><font color="Green" title="E100">A3</font></i></a>, <a class="ref" href="matrix13.html#T96" target="_self" title="MATRIX13:th.96">Th96</a></i>;<br/><b>set </b><font color="Maroon" title="c11">IN</font> = <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c10">N</font>;<br/><b>set </b><font color="Maroon" title="c12">IP</font> = <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c9">P</font>;<br/><b>set </b><font color="Maroon" title="c13">JN</font> = <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c10">N</font>;<br/><b>set </b><font color="Maroon" title="c14">JP</font> = <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c9">P</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:113_1_1_2_1"/>
( ( <font color="Maroon" title="c7">I</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">P</font> ) or ( <font color="Maroon" title="c7">I</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">P</font> ) or ( <font color="Maroon" title="c7">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">P</font> ) or ( <font color="Maroon" title="c7">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">P</font> ) )
 </span><b>by </b><i><a class="txt" href="#E1:113_1_1_2"><i><font color="Green" title="E100">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:113_1_1_2_1_1"/>
( <font color="Maroon" title="c7">I</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span></span><br/><div class="add"><b>then </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> = 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c9">P</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c9">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:113_1_1_2"><i><font color="Green" title="E101">A4</font></i></a>, <a class="txt" href="#E6:113_1_1_2"><i><font color="Green" title="E102">A5</font></i></a>, <a class="txt" href="#E7:113_1_1_2"><i><font color="Green" title="E103">A6</font></i></a>, <a class="ref" href="matrix13.html#T45" target="_self" title="MATRIX13:th.45">Th45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c10">N</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c9">P</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c9">P</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c10">N</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T13" title="MATRIX_9:th.13">MATRIX_9:13</a></i>
;<br/><b>hence </b><a NAME="E3:113_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:113_1_1_2_1_2"/>
( <font color="Maroon" title="c7">I</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span></span><br/><div class="add"><b>then </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> = 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c9">P</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c10">N</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c9">P</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c10">N</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:113_1_1_2"><i><font color="Green" title="E101">A4</font></i></a>, <a class="txt" href="#E6:113_1_1_2"><i><font color="Green" title="E102">A5</font></i></a>, <a class="txt" href="#E7:113_1_1_2"><i><font color="Green" title="E103">A6</font></i></a>, <a class="ref" href="matrix13.html#T45" target="_self" title="MATRIX13:th.45">Th45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c9">P</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c10">N</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c10">N</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c9">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T13" title="MATRIX_9:th.13">MATRIX_9:13</a></i>
;<br/><b>hence </b><a NAME="E3:113_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:113_1_1_2_1_3"/>
( <font color="Maroon" title="c7">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span></span><br/><div class="add"><b>then </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> = 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c9">P</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c10">N</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c9">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:113_1_1_2"><i><font color="Green" title="E101">A4</font></i></a>, <a class="txt" href="#E6:113_1_1_2"><i><font color="Green" title="E102">A5</font></i></a>, <a class="txt" href="#E7:113_1_1_2"><i><font color="Green" title="E103">A6</font></i></a>, <a class="ref" href="matrix13.html#T45" target="_self" title="MATRIX13:th.45">Th45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c10">N</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c9">P</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c9">P</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c10">N</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T13" title="MATRIX_9:th.13">MATRIX_9:13</a></i>
;<br/><b>hence </b><a NAME="E3:113_1_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_2_1_4"><b>suppose </b></a><a NAME="E1:113_1_1_2_1_4"/>
( <font color="Maroon" title="c7">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span></span><br/><div class="add"><b>then </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> = 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c9">P</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c10">N</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c9">P</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c10">N</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:113_1_1_2"><i><font color="Green" title="E101">A4</font></i></a>, <a class="txt" href="#E6:113_1_1_2"><i><font color="Green" title="E102">A5</font></i></a>, <a class="txt" href="#E7:113_1_1_2"><i><font color="Green" title="E103">A6</font></i></a>, <a class="ref" href="matrix13.html#T45" target="_self" title="MATRIX13:th.45">Th45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c9">P</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c10">N</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">J</font>,<font color="Maroon" title="c10">N</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c9">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T13" title="MATRIX_9:th.13">MATRIX_9:13</a></i>
;<br/><b>hence </b><a NAME="E3:113_1_1_2_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</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></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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

<b>assume </b><b>that </b><br/><a NAME="E2:113"/><i><font color="Green" title="E98">A7</font></i>: 
 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>and </b><br/><a NAME="E3:113"/><i><font color="Green" title="E99">A8</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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">i</font>,<font color="Olive" title="b2">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="Olive" title="b3">n</font>,<font color="Olive" title="b4">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> holds <br/><span class="p1">(<span class="default"><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="b3">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="b4">m</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_2"><b>now </b></a><div class="add"><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="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a> <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"> ( <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> &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="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> implies  <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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></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> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:113_2"/><i><font color="Green" title="E100">A9</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="E2:113_2"/><i><font color="Green" title="E101">A10</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></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></span><br/><a NAME="E3:113_2"/><i><font color="Green" title="E102">A11</font></i>: 
(  <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="#E1:113_2"><i><font color="Green" title="E100">A9</font></i></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>;<br/><a NAME="E4:113_2"/><b>then </b><i><font color="Green" title="E103">A12</font></i>: 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<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>,<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>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<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>,<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>
 <b>by </b><i><a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>;<br/><b>set </b><font color="Maroon" title="c7">IN</font> = <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c5">n0</font>;<br/><b>set </b><font color="Maroon" title="c8">IM</font> = <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c6">m0</font>;<br/><b>set </b><font color="Maroon" title="c9">JN</font> = <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">n0</font>;<br/><b>set </b><font color="Maroon" title="c10">JM</font> = <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c6">m0</font>;<br/><a NAME="E5:113_2"/><i><font color="Green" title="E104">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c5">n0</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c6">m0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c6">m0</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">n0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:113"><i><font color="Green" title="E99">A8</font></i></a>, <a class="txt" href="#E2:113_2"><i><font color="Green" title="E101">A10</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:113_2_1"/>
( ( <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j0</font> &amp; <font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m0</font> ) or ( <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j0</font> &amp; <font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">m0</font> ) or ( <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">j0</font> &amp; <font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m0</font> ) or ( <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">j0</font> &amp; <font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">m0</font> ) )
 </span><b>by </b><i><a class="txt" href="#E1:113_2"><i><font color="Green" title="E100">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_2_1_1"><b>suppose </b></a><a NAME="E1:113_2_1_1"/>
( <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j0</font> &amp; <font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></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></span><br/><div class="add"><a NAME="E2:113_2_1_1"/><b>then </b>
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<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>,<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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c5">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c6">m0</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c6">m0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:113_2"><i><font color="Green" title="E103">A12</font></i></a>, <a class="ref" href="matrix13.html#T45" target="_self" title="MATRIX13:th.45">Th45</a></i>;<br/><b>hence </b> <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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c5">n0</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c6">m0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c6">m0</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">n0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:113_2"><i><font color="Green" title="E102">A11</font></i></a>, <a class="ref" href="matrix_9.html#T13" title="MATRIX_9:th.13">MATRIX_9:13</a></i>
<br/>.= 
 <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="#E5:113_2"><i><font color="Green" title="E104">A13</font></i></a>, <a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_2_1_2"><b>suppose </b></a><a NAME="E1:113_2_1_2"/>
( <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j0</font> &amp; <font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">m0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></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></span><br/><div class="add"><a NAME="E2:113_2_1_2"/><b>then </b>
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<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>,<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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c6">m0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c5">n0</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c6">m0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">n0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:113_2"><i><font color="Green" title="E103">A12</font></i></a>, <a class="ref" href="matrix13.html#T45" target="_self" title="MATRIX13:th.45">Th45</a></i>;<br/><b>hence </b> <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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c6">m0</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">n0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c5">n0</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c6">m0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:113_2"><i><font color="Green" title="E102">A11</font></i></a>, <a class="ref" href="matrix_9.html#T13" title="MATRIX_9:th.13">MATRIX_9:13</a></i>
<br/>.= 
 <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="#E5:113_2"><i><font color="Green" title="E104">A13</font></i></a>, <a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_2_1_3"><b>suppose </b></a><a NAME="E1:113_2_1_3"/>
( <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">j0</font> &amp; <font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></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></span><br/><div class="add"><a NAME="E2:113_2_1_3"/><b>then </b>
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<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>,<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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c6">m0</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c5">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c6">m0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:113_2"><i><font color="Green" title="E103">A12</font></i></a>, <a class="ref" href="matrix13.html#T45" target="_self" title="MATRIX13:th.45">Th45</a></i>;<br/><b>hence </b> <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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">n0</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c6">m0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c6">m0</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c5">n0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:113_2"><i><font color="Green" title="E102">A11</font></i></a>, <a class="ref" href="matrix_9.html#T13" title="MATRIX_9:th.13">MATRIX_9:13</a></i>
<br/>.= 
 <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="#E5:113_2"><i><font color="Green" title="E104">A13</font></i></a>, <a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_2_1_4"><b>suppose </b></a><a NAME="E1:113_2_1_4"/>
( <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">j0</font> &amp; <font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">m0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></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></span><br/><div class="add"><a NAME="E2:113_2_1_4"/><b>then </b>
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<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>,<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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c6">m0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">n0</font></span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c6">m0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c5">n0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:113_2"><i><font color="Green" title="E103">A12</font></i></a>, <a class="ref" href="matrix13.html#T45" target="_self" title="MATRIX13:th.45">Th45</a></i>;<br/><b>hence </b> <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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c6">m0</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c5">n0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">n0</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i0</font>,<font color="Maroon" title="c6">m0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:113_2"><i><font color="Green" title="E102">A11</font></i></a>, <a class="ref" href="matrix_9.html#T13" title="MATRIX_9:th.13">MATRIX_9:13</a></i>
<br/>.= 
 <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="#E5:113_2"><i><font color="Green" title="E104">A13</font></i></a>, <a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:113"/>
 <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
 <b>by </b><i><a class="txt" href="#E2:113"><i><font color="Green" title="E98">A7</font></i></a>, <a class="ref" href="matrix13.html#T96" target="_self" title="MATRIX13:th.96">Th96</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>
