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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>] <span class="kw">means </span>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,$1)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2;<br/>
<a NAME="E1:15_1_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="goboard1.html#E10"><span class="lab"><font color="Green" title="E10">Lm1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:15_1_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">m</font> = <font color="Maroon" title="c3">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E5:15_1_1"/><span class="lab"><font color="Green" title="E17">A4</font></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> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>
 
<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="E6:15_1_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c1">G</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c5">k</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> <font color="Maroon" title="c1">G</font> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_1_1"/>
<font color="Maroon" title="c5">k</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> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font></span><br/>

<a NAME="E2:15_1_1_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">k</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> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E3:15_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">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#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:15_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T104" title="FINSEQ_3:th.104">FINSEQ_3:104</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:15_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:15_1_1_1"><span class="lab"><font color="Green" title="E19">A6</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="E7:15_1_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
 for <font color="Olive" title="b1">k</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">k</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> <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="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> ex <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c6">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="c5">k</font>,<font color="Maroon" title="c6">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> <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="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies  ex <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">j</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">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> <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="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">j</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:15_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A9</font></span>: 
( <font color="Maroon" title="c5">k</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> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c6">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> <font color="Maroon" title="c4">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> =  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>) as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T105" title="FINSEQ_3:th.105">FINSEQ_3:105</a></span>;<br/>
<a NAME="E4:15_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E2:15_1_1_2"><span class="lab"><font color="Green" title="E20">A9</font></span></a></span>;<br/>
<a NAME="E5:15_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">j</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> <font color="Maroon" title="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_2"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">x</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">x</font>]</span><br/>

<span class="kw">thus </span><a NAME="E7:15_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">x</font>]
 ; <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>
<span class="kw">consider </span><font color="Maroon" title="c5">N</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">m</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E9:15_1_1"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
 for <font color="Olive" title="b1">k</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">k</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="c5">N</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">j</font>)]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="matrix_1.html#S2" title="MATRIX_1:sch.2">MATRIX_1:sch 2</a>(<a class="txt" href="#E7:15_1_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>);<br/></span>
<a NAME="E10:15_1_1"/><span class="lab"><font color="Green" title="E21">A11</font></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> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">N</font>
 
<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="E11:15_1_1"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">N</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<a NAME="E12:15_1_1"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
 for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b2">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> <font color="Maroon" title="c4">m</font> holds <br/><font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">k</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="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c6">k</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> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c7">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> <font color="Maroon" title="c4">m</font> implies <font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:15_1_1_3"/>
( <font color="Maroon" title="c6">k</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> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c7">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> <font color="Maroon" title="c4">m</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="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span><br/>

<a NAME="E2:15_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">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="c5">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:15_1_1_3"/>
<font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15_1_1"><span class="lab"><font color="Green" title="E20">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="E13:15_1_1"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
 for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b2">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> <font color="Maroon" title="c4">m</font> &amp;  not <font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">k</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">k</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c6">k</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> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c7">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> <font color="Maroon" title="c4">m</font> &amp;  not <font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> implies <font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:15_1_1_4"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
( <font color="Maroon" title="c6">k</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> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c7">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> <font color="Maroon" title="c4">m</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="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<a NAME="E2:15_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:15_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a></span>;<br/>
<a NAME="E3:15_1_1_4"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E4:15_1_1_4"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E5:15_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c2">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#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</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">hence </span><a NAME="E6:15_1_1_4"/>
( <font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_4"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E2:15_1_1_4"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E3:15_1_1_4"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="seqm_3.html#T44" title="SEQM_3:th.44">SEQM_3:44</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="E14:15_1_1"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive" title="b1">k</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> <font color="Maroon" title="c4">m</font> or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Olive" title="b1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">k</font>) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Olive" title="b1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">k</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> <font color="Maroon" title="c4">m</font> or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_1_5"/><span class="lab"><font color="Green" title="E26">A19</font></span>: 
<font color="Maroon" title="c6">k</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> <font color="Maroon" title="c4">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )</span><br/>

<a NAME="E2:15_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A20</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E3:15_1_1_5"/>
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E4:15_1_1_5"/><span class="kw">then </span>
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_5"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:15_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A21</font></span>: 
( <font color="Maroon" title="c6">k</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> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:15_1_1_5"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E6:15_1_1_5"/><span class="lab"><font color="Green" title="E29">A22</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">N</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_1_1_5_1_1"/>
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font> or <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_5_1_1_1"/><span class="lab"><font color="Green" title="E30">A23</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_5_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:15_1_1_5_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</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#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span><br/><a NAME="E2:15_1_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A24</font></span>: 
<font color="Maroon" title="c7">j</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> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1_5"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
<font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c6">k</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">j</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E12:15_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E1:15_1_1_5"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E2:15_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E31">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_5_1_1_1"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="finseq_3.html#T110" title="FINSEQ_3:th.110">FINSEQ_3:110</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E5:15_1_1_5"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="txt" href="#E2:15_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="ref" href="goboard1.html#T2" target="_self" title="GOBOARD1:th.2">Th17</a></span>
;<br/> <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><span class="kw">hence </span><a NAME="E3:15_1_1_5_1_1_1"/>
(  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E6:15_1_1_5"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_5_1_1_2"/><span class="lab"><font color="Green" title="E30">A25</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_5_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:15_1_1_5_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</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#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span><br/><a NAME="E2:15_1_1_5_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A26</font></span>: 
<font color="Maroon" title="c7">j</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> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1_5"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:15_1_1_5_1_1_2_1"/><span class="lab"><font color="Green" title="E32">A27</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E4:15_1_1_5_1_1_2_1"/><span class="lab"><font color="Green" title="E33">A28</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">j</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/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
<font color="Maroon" title="c5">N</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c6">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">j</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E12:15_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E1:15_1_1_5"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E2:15_1_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E31">A26</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">j</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:15_1_1_5"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="txt" href="#E1:15_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="txt" href="#E3:15_1_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E4:15_1_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="ref" href="finseq_3.html#T111" title="FINSEQ_3:th.111">FINSEQ_3:111</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E5:15_1_1_5"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="txt" href="#E2:15_1_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="ref" href="goboard1.html#T2" target="_self" title="GOBOARD1:th.2">Th17</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><span class="kw">hence </span><a NAME="E3:15_1_1_5_1_1_2"/>
(  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E6:15_1_1_5"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:15_1_1_5"/>
(  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">k</font>) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</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>
<span class="kw">reconsider </span><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c5">N</font> as   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E16:15_1_1"/>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E17:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A29</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c7">k</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> <font color="Maroon" title="c6">M</font> implies  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:15_1_1_6"/><span class="lab"><font color="Green" title="E28">A31</font></span>: 
<font color="Maroon" title="c7">k</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> <font color="Maroon" title="c6">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> </span><br/><a NAME="E2:15_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A32</font></span>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="goboard1.html#D4" target="_self" title="GOBOARD1:def.4">Def6</a></span>;<br/><a NAME="E3:15_1_1_6"/>
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E4:15_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A33</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">m</font> <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="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>), <font color="Maroon" title="c9">lg</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">k</font>) as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c10">X</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font>;<br/><span class="kw">set </span><font color="Maroon" title="c11">xg</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c9">lg</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c12">n</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="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> &amp; <font color="Maroon" title="c13">j</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="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> implies <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:15_1_1_6_1"/><span class="lab"><font color="Green" title="E31">A34</font></span>: 
( <font color="Maroon" title="c12">n</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="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> &amp; <font color="Maroon" title="c13">j</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="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</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="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span><br/><a NAME="E2:15_1_1_6_1"/><span class="lab"><font color="Green" title="E32">A35</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lg</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D1" target="_self" title="GOBOARD1:def.1">Def3</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E3:15_1_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A36</font></span>: 
( <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>) &amp; <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>) &amp; <font color="Maroon" title="c12">n</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> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c13">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> <font color="Maroon" title="c4">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E1:15_1_1_6_1"><span class="lab"><font color="Green" title="E31">A34</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E4:15_1_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A37</font></span>: 
( ( <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> or <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) &amp; ( <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> or <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E13:15_1_1"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E1:15_1_1_6"><span class="lab"><font color="Green" title="E28">A31</font></span></a></span>;<br/><a NAME="E5:15_1_1_6_1"/>
( <font color="Maroon" title="c12">n</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> <font color="Maroon" title="c8">L</font> &amp; <font color="Maroon" title="c13">j</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> <font color="Maroon" title="c8">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_6_1"><span class="lab"><font color="Green" title="E31">A34</font></span></a>, <a class="txt" href="#E2:15_1_1_6_1"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:15_1_1_6_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E7:15_1_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A38</font></span>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_6_1"><span class="lab"><font color="Green" title="E31">A34</font></span></a>, <a class="txt" href="#E3:15_1_1_6_1"><span class="lab"><font color="Green" title="E33">A36</font></span></a>, <a class="ref" href="goboard1.html#D1" target="_self" title="GOBOARD1:def.1">Def3</a></span>;<br/><a NAME="E8:15_1_1_6_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E1:15_1_1_6_1"><span class="lab"><font color="Green" title="E31">A34</font></span></a>, <a class="txt" href="#E2:15_1_1_6_1"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E9:15_1_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A39</font></span>: 
( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E10:15_1_1_6_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E11:15_1_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A40</font></span>: 
( <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c12">n</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> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c13">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> <font color="Maroon" title="c1">G</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E4:15_1_1_6"><span class="lab"><font color="Green" title="E30">A33</font></span></a>, <a class="txt" href="#E1:15_1_1_6_1"><span class="lab"><font color="Green" title="E31">A34</font></span></a>, <a class="txt" href="#E2:15_1_1_6_1"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="txt" href="#E9:15_1_1_6_1"><span class="lab"><font color="Green" title="E36">A39</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E12:15_1_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A41</font></span>: 
( <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>) &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>) &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E13:15_1_1_6_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_6_1"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E14:15_1_1_6_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_6_1"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="txt" href="#E11:15_1_1_6_1"><span class="lab"><font color="Green" title="E37">A40</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E15:15_1_1_6_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_6_1"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="txt" href="#E11:15_1_1_6_1"><span class="lab"><font color="Green" title="E37">A40</font></span></a>, <a class="txt" href="#E12:15_1_1_6_1"><span class="lab"><font color="Green" title="E38">A41</font></span></a>, <a class="ref" href="goboard1.html#D1" target="_self" title="GOBOARD1:def.1">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E16:15_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_6"><span class="lab"><font color="Green" title="E29">A32</font></span></a>, <a class="txt" href="#E2:15_1_1_6_1"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="txt" href="#E3:15_1_1_6_1"><span class="lab"><font color="Green" title="E33">A36</font></span></a>, <a class="txt" href="#E4:15_1_1_6_1"><span class="lab"><font color="Green" title="E34">A37</font></span></a>, <a class="txt" href="#E7:15_1_1_6_1"><span class="lab"><font color="Green" title="E35">A38</font></span></a>, <a class="txt" href="#E11:15_1_1_6_1"><span class="lab"><font color="Green" title="E37">A40</font></span></a>, <a class="txt" href="#E12:15_1_1_6_1"><span class="lab"><font color="Green" title="E38">A41</font></span></a>, <a class="ref" href="seqm_3.html#D10" title="SEQM_3:def.10">SEQM_3:def 10</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><span class="kw">hence </span><a NAME="E7:15_1_1_6"/>
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#D10" title="SEQM_3:def.10">SEQM_3:def 10</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>
<div><span class="lab"><font color="Green" title="E28">A42</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c7">k</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> <font color="Maroon" title="c6">M</font></span>)</span> implies  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:15_1_1_7"/><span class="lab"><font color="Green" title="E29">A43</font></span>: 
<font color="Maroon" title="c7">k</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> <font color="Maroon" title="c6">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> </span><br/><a NAME="E2:15_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A44</font></span>: 
(  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">k</font>) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E14:15_1_1"><span class="lab"><font color="Green" title="E25">A18</font></span></a></span>;<br/><a NAME="E3:15_1_1_7"/><span class="lab"><font color="Green" title="E31">A45</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E1:15_1_1_7"><span class="lab"><font color="Green" title="E29">A43</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:15_1_1_7"/>
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E5:15_1_1_7"/><span class="kw">then </span>
( <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1_7"><span class="lab"><font color="Green" title="E31">A45</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:15_1_1_7"/><span class="kw">then </span>
( <font color="Maroon" title="c7">k</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> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:15_1_1_7"><span class="lab"><font color="Green" title="E31">A45</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:15_1_1_7"/>
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_7"><span class="lab"><font color="Green" title="E30">A44</font></span></a>, <a class="ref" href="goboard1.html#D5" target="_self" title="GOBOARD1:def.5">Def7</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>
<div><span class="lab"><font color="Green" title="E29">A46</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c7">k</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> <font color="Maroon" title="c6">M</font> implies  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>), <font color="Maroon" title="c9">lg</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">k</font>) as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c10">X</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font>;<br/><span class="kw">set </span><font color="Maroon" title="c11">xg</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c9">lg</font>;<br/><a NAME="E2:15_1_1_8"/>
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:15_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A47</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">m</font> <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="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/><span class="kw">assume </span><a NAME="E4:15_1_1_8"/><span class="lab"><font color="Green" title="E31">A48</font></span>: 
<font color="Maroon" title="c7">k</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> <font color="Maroon" title="c6">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> </span><br/><a NAME="E5:15_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A49</font></span>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c9">lg</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="goboard1.html#D6" target="_self" title="GOBOARD1:def.6">Def8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_8_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c12">n</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="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> &amp; <font color="Maroon" title="c13">j</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="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> implies <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:15_1_1_8_1"/><span class="lab"><font color="Green" title="E33">A50</font></span>: 
( <font color="Maroon" title="c12">n</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="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> &amp; <font color="Maroon" title="c13">j</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="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> )
 ; <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="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></span><br/><a NAME="E2:15_1_1_8_1"/><span class="lab"><font color="Green" title="E34">A51</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lg</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D2" target="_self" title="GOBOARD1:def.2">Def4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E3:15_1_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A52</font></span>: 
( <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>) &amp; <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>) &amp; <font color="Maroon" title="c12">n</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> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c13">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> <font color="Maroon" title="c4">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E4:15_1_1_8_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E5:15_1_1_8_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c8">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E6:15_1_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A53</font></span>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E3:15_1_1_8_1"><span class="lab"><font color="Green" title="E35">A52</font></span></a>, <a class="ref" href="goboard1.html#D2" target="_self" title="GOBOARD1:def.2">Def4</a></span>;<br/><a NAME="E7:15_1_1_8_1"/><span class="lab"><font color="Green" title="E37">A54</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:15_1_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A55</font></span>: 
( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E9:15_1_1_8_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E10:15_1_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A56</font></span>: 
( <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c12">n</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> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c13">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> <font color="Maroon" title="c1">G</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E3:15_1_1_8"><span class="lab"><font color="Green" title="E30">A47</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E8:15_1_1_8_1"><span class="lab"><font color="Green" title="E38">A55</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E11:15_1_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A57</font></span>: 
( <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>) &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>) &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E12:15_1_1_8_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E13:15_1_1_8_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c9">lg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lg</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E10:15_1_1_8_1"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E14:15_1_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A58</font></span>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c9">lg</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E10:15_1_1_8_1"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="txt" href="#E11:15_1_1_8_1"><span class="lab"><font color="Green" title="E40">A57</font></span></a>, <a class="ref" href="goboard1.html#D2" target="_self" title="GOBOARD1:def.2">Def4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">r</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c15">s</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font>;<br/><a NAME="E15:15_1_1_8_1"/><span class="lab"><font color="Green" title="E42">A59</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c9">lg</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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_8_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_1_1_8_1_1"/>
( <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font> or <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_8_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_8_1_1_1"/><span class="lab"><font color="Green" title="E43">A60</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font>
 ; <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="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:15_1_1_8_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A61</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="lab"><font color="Green" title="E45">A62</font></span>: <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>) = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c9">lg</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E12:15_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E4:15_1_1_8"><span class="lab"><font color="Green" title="E31">A48</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1_8_1"><span class="lab"><font color="Green" title="E40">A57</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1_1_1"><span class="lab"><font color="Green" title="E44">A61</font></span></a>, <a class="ref" href="finseq_3.html#T110" title="FINSEQ_3:th.110">FINSEQ_3:110</a></span>
;<br/><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>) = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c9">lg</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E12:15_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E4:15_1_1_8"><span class="lab"><font color="Green" title="E31">A48</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1_8_1"><span class="lab"><font color="Green" title="E40">A57</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1_1_1"><span class="lab"><font color="Green" title="E43">A60</font></span></a>, <a class="ref" href="finseq_3.html#T110" title="FINSEQ_3:th.110">FINSEQ_3:110</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:15_1_1_8_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E3:15_1_1_8"><span class="lab"><font color="Green" title="E30">A47</font></span></a>, <a class="txt" href="#E5:15_1_1_8"><span class="lab"><font color="Green" title="E32">A49</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E6:15_1_1_8_1"><span class="lab"><font color="Green" title="E36">A53</font></span></a>, <a class="txt" href="#E14:15_1_1_8_1"><span class="lab"><font color="Green" title="E41">A58</font></span></a>, <a class="txt" href="#E3:15_1_1_8_1_1_1"><span class="lab"><font color="Green" title="E45">A62</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_8_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_8_1_1_2"/><span class="lab"><font color="Green" title="E43">A63</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">i</font>
 ; <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="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E44">A64</font></span>: <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c13">j</font>) = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c9">lg</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E12:15_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E4:15_1_1_8"><span class="lab"><font color="Green" title="E31">A48</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E7:15_1_1_8_1"><span class="lab"><font color="Green" title="E37">A54</font></span></a>, <a class="txt" href="#E11:15_1_1_8_1"><span class="lab"><font color="Green" title="E40">A57</font></span></a>, <a class="txt" href="#E15:15_1_1_8_1"><span class="lab"><font color="Green" title="E42">A59</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1_1_2"><span class="lab"><font color="Green" title="E43">A63</font></span></a>, <a class="ref" href="finseq_3.html#T111" title="FINSEQ_3:th.111">FINSEQ_3:111</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_8_1_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_8_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_1_1_8_1_1_2_2_1"/>
( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font> or <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_8_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_8_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E45">A65</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font>
 ; <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="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span><br/><div class="add"><a NAME="E2:15_1_1_8_1_1_2_2_1_1"/>
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:15_1_1_8_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A66</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>) = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c9">lg</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E12:15_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E4:15_1_1_8"><span class="lab"><font color="Green" title="E31">A48</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1_8_1"><span class="lab"><font color="Green" title="E40">A57</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E45">A65</font></span></a>, <a class="ref" href="finseq_3.html#T110" title="FINSEQ_3:th.110">FINSEQ_3:110</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:15_1_1_8_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1_8"><span class="lab"><font color="Green" title="E32">A49</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E6:15_1_1_8_1"><span class="lab"><font color="Green" title="E36">A53</font></span></a>, <a class="txt" href="#E10:15_1_1_8_1"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="txt" href="#E14:15_1_1_8_1"><span class="lab"><font color="Green" title="E41">A58</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1_1_2"><span class="lab"><font color="Green" title="E44">A64</font></span></a>, <a class="txt" href="#E3:15_1_1_8_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E46">A66</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_8_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_8_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E45">A67</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">i</font>
 ; <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="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span><br/><div class="add"><a NAME="E2:15_1_1_8_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E46">A68</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c12">n</font>) = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c9">lg</font>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E12:15_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E4:15_1_1_8"><span class="lab"><font color="Green" title="E31">A48</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E7:15_1_1_8_1"><span class="lab"><font color="Green" title="E37">A54</font></span></a>, <a class="txt" href="#E11:15_1_1_8_1"><span class="lab"><font color="Green" title="E40">A57</font></span></a>, <a class="txt" href="#E15:15_1_1_8_1"><span class="lab"><font color="Green" title="E42">A59</font></span></a>, <a class="txt" href="#E1:15_1_1_8_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E45">A67</font></span></a>, <a class="ref" href="finseq_3.html#T111" title="FINSEQ_3:th.111">FINSEQ_3:111</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:15_1_1_8_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1_8"><span class="lab"><font color="Green" title="E32">A49</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E6:15_1_1_8_1"><span class="lab"><font color="Green" title="E36">A53</font></span></a>, <a class="txt" href="#E10:15_1_1_8_1"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="txt" href="#E14:15_1_1_8_1"><span class="lab"><font color="Green" title="E41">A58</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1_1_2"><span class="lab"><font color="Green" title="E44">A64</font></span></a>, <a class="txt" href="#E2:15_1_1_8_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E46">A68</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:15_1_1_8_1_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 ; <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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:15_1_1_8"/>
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c7">k</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> <font color="Maroon" title="c6">M</font></span>)</span> implies  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:15_1_1_9"/><span class="lab"><font color="Green" title="E30">A69</font></span>: 
<font color="Maroon" title="c7">k</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> <font color="Maroon" title="c6">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> </span><br/><a NAME="E2:15_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A70</font></span>: 
(  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">k</font>) or  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E14:15_1_1"><span class="lab"><font color="Green" title="E25">A18</font></span></a></span>;<br/><a NAME="E3:15_1_1_9"/><span class="lab"><font color="Green" title="E32">A71</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E1:15_1_1_9"><span class="lab"><font color="Green" title="E30">A69</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:15_1_1_9"/>
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E5:15_1_1_9"/><span class="kw">then </span>
( <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1_9"><span class="lab"><font color="Green" title="E32">A71</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:15_1_1_9"/><span class="kw">then </span>
( <font color="Maroon" title="c7">k</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> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:15_1_1_9"><span class="lab"><font color="Green" title="E32">A71</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:15_1_1_9"/>
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c7">k</font>)</span>)</span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_9"><span class="lab"><font color="Green" title="E31">A70</font></span></a>, <a class="ref" href="goboard1.html#D7" target="_self" title="GOBOARD1:def.7">Def9</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">M</font> = <font color="Maroon" title="c6">M</font> as  <a href="relat_1.html#V3" title="RELAT_1:attr.3">V3</a>()  <a href="goboard1.html#V2" title="GOBOARD1:attr.2">X_equal-in-line</a>   <a href="goboard1.html#V3" title="GOBOARD1:attr.3">Y_equal-in-column</a>   <a href="goboard1.html#V4" title="GOBOARD1:attr.4">Y_increasing-in-line</a>   <a href="goboard1.html#V5" title="GOBOARD1:attr.5">X_increasing-in-column</a>  <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15_1_1"><span class="lab"><font color="Green" title="E26">A29</font></span></a>, <a class="txt" href="#E18:15_1_1"><span class="lab"><font color="Green" title="E27">A30</font></span></a>, <a class="txt" href="#E19:15_1_1"><span class="lab"><font color="Green" title="E28">A42</font></span></a>, <a class="txt" href="#E20:15_1_1"><span class="lab"><font color="Green" title="E29">A46</font></span></a>, <a class="ref" href="goboard1.html#D3" target="_self" title="GOBOARD1:def.3">Def5</a>, <a class="ref" href="goboard1.html#D4" target="_self" title="GOBOARD1:def.4">Def6</a>, <a class="ref" href="goboard1.html#D5" target="_self" title="GOBOARD1:def.5">Def7</a>, <a class="ref" href="goboard1.html#D6" target="_self" title="GOBOARD1:def.6">Def8</a>, <a class="ref" href="goboard1.html#D7" target="_self" title="GOBOARD1:def.7">Def9</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">M</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c1">G</font> holds <br/><font color="Maroon" title="c7">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E23:15_1_1"/><span class="lab"><font color="Green" title="E30">A72</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>; <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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c1">G</font> holds <br/><font color="Maroon" title="c7">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c8">k</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> <font color="Maroon" title="c1">G</font> implies <font color="Maroon" title="c7">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c8">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E24:15_1_1"/><span class="lab"><font color="Green" title="E31">A73</font></span>: 
<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c1">G</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="c7">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c8">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span><br/>

<a NAME="E25:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A74</font></span>: 
<font color="Maroon" title="c7">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E23:15_1_1"><span class="lab"><font color="Green" title="E30">A72</font></span></a>, <a class="ref" href="matrix_2.html#T16" title="MATRIX_2:th.16">MATRIX_2:16</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">s</font> = <font color="Maroon" title="c7">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E27:15_1_1"/><span class="lab"><font color="Green" title="E33">A75</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E25:15_1_1"><span class="lab"><font color="Green" title="E32">A74</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E28:15_1_1"/><span class="lab"><font color="Green" title="E34">A76</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c8">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E24:15_1_1"><span class="lab"><font color="Green" title="E31">A73</font></span></a></span>;<br/>
<a NAME="E29:15_1_1"/><span class="lab"><font color="Green" title="E35">A77</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:15_1_1"><span class="lab"><font color="Green" title="E33">A75</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">j</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> <font color="Maroon" title="c9">s</font> implies <font color="Maroon" title="c9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c8">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:15_1_1_10"/><span class="lab"><font color="Green" title="E36">A78</font></span>: 
<font color="Maroon" title="c10">j</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> <font color="Maroon" title="c9">s</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="c9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c8">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span><br/><a NAME="E2:15_1_1_10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c8">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c10">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E29:15_1_1"><span class="lab"><font color="Green" title="E35">A77</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:15_1_1_10"/>
<font color="Maroon" title="c9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c8">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:15_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E24:15_1_1"><span class="lab"><font color="Green" title="E31">A73</font></span></a>, <a class="txt" href="#E25:15_1_1"><span class="lab"><font color="Green" title="E32">A74</font></span></a>, <a class="txt" href="#E1:15_1_1_10"><span class="lab"><font color="Green" title="E36">A78</font></span></a>, <a class="txt" href="#E29:15_1_1"><span class="lab"><font color="Green" title="E35">A77</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>
<span class="kw">hence </span><a NAME="E31:15_1_1"/>
<font color="Maroon" title="c7">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c8">k</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:15_1_1"><span class="lab"><font color="Green" title="E33">A75</font></span></a>, <a class="txt" href="#E28:15_1_1"><span class="lab"><font color="Green" title="E34">A76</font></span></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</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>
