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<span class="kw">let </span><font color="Maroon" title="c1">l</font>, <font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font>  st <font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Olive" title="b2">a</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="Olive" title="b1">K</font> holds <br/>(  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Olive" title="b2">a</font>) is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Olive" title="b2">a</font>)</span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font>  st <font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Olive" title="b1">a</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="c3">K</font> holds <br/>(  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Olive" title="b1">a</font>) is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Olive" title="b1">a</font>)</span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c4">a</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="c3">K</font> implies (  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>) is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>) ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:24"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c4">a</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="c3">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>) is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">A</font> =  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>);<br/>
<span class="lab"><font color="Green" title="E22">A3</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> = 
 <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> <span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <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> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B</font> =  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>);<br/>
<span class="lab"><font color="Green" title="E23">A4</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> = 
 <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> <span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <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> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E5:24"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix12.html#T1" target="_self" title="MATRIX12:th.1">Th1</a></span>;<br/>
<a NAME="E6:24"/>
 for <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 <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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <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#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <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>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:24_3"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>)</span><br/>

<a NAME="E2:24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E3:24_3"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_3"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E4:24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A9</font></span>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></span>;<br/>
<a NAME="E5:24_3"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E3:24_3"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E29">A10</font></span>: <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>) = 
<font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E3:24_3"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="vectsp_2.html#D2" title="VECTSP_2:def.2">VECTSP_2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
;<br/>
<a NAME="E7:24_3"/>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E3:24_3"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:24_3"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_3"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E6:24_3"><span class="lab"><font color="Green" title="E29">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A11</font></span>: 
 <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></span>;<br/>
<a NAME="E8:24"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix12.html#T1" target="_self" title="MATRIX12:th.1">Th1</a></span>;<br/>
<a NAME="E9:24"/>
 for <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 <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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <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#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <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>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:24_4"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>)</span><br/>

<a NAME="E2:24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E3:24_4"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_4"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E4:24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A16</font></span>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_4"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></span>;<br/>
<a NAME="E5:24_4"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E3:24_4"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E31">A17</font></span>: <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>) = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E3:24_4"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="vectsp_2.html#D2" title="VECTSP_2:def.2">VECTSP_2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c8">j</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
;<br/>
<a NAME="E7:24_4"/>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E2:24_4"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E3:24_4"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:24_4"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_4"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E6:24_4"><span class="lab"><font color="Green" title="E31">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A18</font></span>: 
 <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></span>;<br/>
<a NAME="E11:24"/>
( <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>) &amp; <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>)</span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="matrix12.html#T7" target="_self" title="MATRIX12:th.7">Th7</a></span>;<br/>
<a NAME="E12:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A19</font></span>: 
 <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>) <a href="matrix_6.html#R2" title="MATRIX_6:pred.2">is_reverse_of</a>  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="matrix_6.html#D2" title="MATRIX_6:def.2">MATRIX_6:def 2</a></span>;<br/>
<a NAME="E13:24"/><span class="kw">then </span>
 <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>) is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_6.html#D3" title="MATRIX_6:def.3">MATRIX_6:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:24"/>
(  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>) is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c4">a</font>)</span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">l</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="ref" href="matrix_6.html#D4" title="MATRIX_6:def.4">MATRIX_6:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
