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<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/>









<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E43</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green">E44</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> )
 <b>and </b><br/><a NAME="E4:4"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E5:4"/>
<a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>;<br/>
<b>then </b><i><font color="Green">E46</font></i>:  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> = 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
 <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E7:4"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X1</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="pscomp_1.html#T64">PSCOMP_1:64</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:4"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K25">E-most</a> <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E51</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T108">PSCOMP_1:108</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<a NAME="E13:4"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:4"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:4"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:4"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">k1</font></span>)</span> where <font color="Olive">k1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k1</font> &amp; <font color="Olive">k1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D6">TOPREAL1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:4"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E19:4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">k1</font></span>)</span> where <font color="Olive">k1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k1</font> &amp; <font color="Olive">k1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E21:4"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>
 <b>and </b><br/><a NAME="E22:4"/><i><font color="Green">E59</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>and </b><br/><a NAME="E23:4"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>
<a NAME="E24:4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E25:4"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E26:4"/><b>then </b>
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E27:4"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E28:4"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">f</font> is <a href="topreal1.html#V1">special</a>
 
<b>by </b><i>, <a class="ref" href="jordan8.html#T7">JORDAN8:7</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E29:4"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E30:4"/><b>then </b>
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E31:4"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">j0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E33:4"/><i><font color="Green">E67</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j0</font>,<font color="Maroon">i0</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,<font color="Maroon">i0</font> )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">jo</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">io</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E35:4"/><i><font color="Green">E70</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">jo</font>,<font color="Maroon">io</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,<font color="Maroon">io</font> )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>
<a NAME="E36:4"/><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E37:4"/><i><font color="Green">E72</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E38:4"/><i><font color="Green">E73</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j0</font> &amp; <font color="Maroon">j0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E39:4"/><i><font color="Green">E74</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jo</font> &amp; <font color="Maroon">jo</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">io</font> &amp; <font color="Maroon">io</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/>
<a NAME="E41:4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E42:4"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , , <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/>
<a NAME="E43:4"/><b>then </b><i><font color="Green">E77</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/>
<a NAME="E44:4"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 </span><b>by </b><i>, , , <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1"><b>suppose </b></a><a NAME="E1:4_4_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i0</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">i0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,<font color="Maroon">i0</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
;<br/><a NAME="E3:4_4_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/><a NAME="E4:4_4_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">io</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">io</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,<font color="Maroon">io</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
;<br/><a NAME="E6:4_4_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">io</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">io</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/><a NAME="E7:4_4_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">io</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>thus </b><a NAME="E8:4_4_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_1_1_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:4_4_1_1_3_1_1"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:4_4_1_1_3_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1_1_3_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_1_1_3_1_1_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> or  not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:4_4_1_1_3_1_1_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:4_4_1_1_3_1_1_1_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:4_4_1_1_3_1_1_1_1_1"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:4_4_1_1_3_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a></i>;<br/><a NAME="E5:4_4_1_1_3_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:4_4_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/><a NAME="E7:4_4_1_1_3_1_1_1_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><a NAME="E8:4_4_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">jo</font>;<br/><a NAME="E9:4_4_1_1_3_1_1_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green">E88</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><i><font color="Green">E89</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/><a NAME="E12:4_4_1_1_3_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><a NAME="E13:4_4_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><b>hence </b><a NAME="E14:4_4_1_1_3_1_1_1_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>thus </b><a NAME="E15:4_4_1_1_3_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="jordan1f.html#T1" target="_self">Th1</a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>thus </b><a NAME="E16:4_4_1_1_3_1_1_1_1_1"/>
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:4_4_1_1_3_1_1_1_1_2"/><i><font color="Green">E91</font></i>: 
not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><div class="add"><i><font color="Green">E92</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
;<br/><b>thus </b><a NAME="E3:4_4_1_1_3_1_1_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_1_1_3_1_1_1_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 </span><b>by </b><i>, , , <a class="ref" href="goboard7.html#T9">GOBOARD7:9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:4_4_1_1_3_1_1_1_1_2_2_1_1"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
<font color="Maroon">jo</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , , , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><a NAME="E4:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">jo</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
<font color="Maroon">j0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jo</font> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, , , , , , , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><a NAME="E6:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">j0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jo</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E7:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">j0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jo</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E8:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">io</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jo</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , , , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E9:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jo</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/><a NAME="E10:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">j0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jo</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/><a NAME="E11:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">jo</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j0</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E12:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">j0</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">jo</font> <a href="nat_1.html#K1">+</a> 0
 ;<br/><a NAME="E13:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">j0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,<font color="Maroon">i0</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><a NAME="E15:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E16:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E17:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j0</font>
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><a NAME="E19:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j0</font>
 <b>by </b><i><a class="ref" href="jordan1f.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">j0</font>;<br/><i><font color="Green">E99</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><a NAME="E21:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,<font color="Maroon">i0</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><a NAME="E22:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E23:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E24:4_4_1_1_3_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E25:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="txt" href="jordan1f.html#E1:4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:4_4_1_1_3_1_1_1_1_2_2_1_2"/><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:4_4_1_1_3_1_1_1_1_2_2_1_2"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><b>hence </b><a NAME="E3:4_4_1_1_3_1_1_1_1_2_2_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:4_4_1_1_3_1_2"/><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:4_4_1_1_3_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1_1_3_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_1_1_3_1_2_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> or  not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:4_4_1_1_3_1_2_1_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:4_4_1_1_3_1_2_1_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:4_4_1_1_3_1_2_1_1_1"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:4_4_1_1_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a></i>;<br/><a NAME="E5:4_4_1_1_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:4_4_1_1_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/><a NAME="E7:4_4_1_1_3_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><a NAME="E8:4_4_1_1_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">j0</font>;<br/><a NAME="E9:4_4_1_1_3_1_2_1_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green">E106</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><i><font color="Green">E107</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/><a NAME="E12:4_4_1_1_3_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><a NAME="E13:4_4_1_1_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><b>hence </b><a NAME="E14:4_4_1_1_3_1_2_1_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>thus </b><a NAME="E15:4_4_1_1_3_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="jordan1f.html#T1" target="_self">Th1</a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>thus </b><a NAME="E16:4_4_1_1_3_1_2_1_1_1"/>
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:4_4_1_1_3_1_2_1_1_2"/><i><font color="Green">E109</font></i>: 
not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><div class="add"><i><font color="Green">E110</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
;<br/><b>thus </b><a NAME="E3:4_4_1_1_3_1_2_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_4_1_1_3_1_2_1_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 </span><b>by </b><i>, , , <a class="ref" href="goboard7.html#T9">GOBOARD7:9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:4_4_1_1_3_1_2_1_1_2_2_1_1"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
<font color="Maroon">j0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , , , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><a NAME="E4:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">j0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
<font color="Maroon">jo</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j0</font> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, , , , , , , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><a NAME="E6:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">jo</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j0</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E7:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">jo</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j0</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E8:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">io</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jo</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , , , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E9:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jo</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/><a NAME="E10:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">j0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jo</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E11:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">jo</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j0</font> <a href="xcmplx_0.html#K6">-</a> 0
 ;<br/><a NAME="E12:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">jo</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,<font color="Maroon">io</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><a NAME="E14:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E15:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E16:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jo</font>
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><a NAME="E18:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jo</font>
 <b>by </b><i><a class="ref" href="jordan1f.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">jo</font>;<br/><i><font color="Green">E117</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><a NAME="E20:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,<font color="Maroon">io</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><a NAME="E21:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E22:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">jo</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E23:4_4_1_1_3_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E24:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="txt" href="jordan1f.html#E1:4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:4_4_1_1_3_1_2_1_1_2_2_1_2"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:4_4_1_1_3_1_2_1_1_2_2_1_2"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><b>hence </b><a NAME="E3:4_4_1_1_3_1_2_1_1_2_2_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

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

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_2"><b>suppose </b></a><a NAME="E1:4_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:4_4_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">j0</font>;<br/><i><font color="Green">E120</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><i><font color="Green">E121</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j0</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/><a NAME="E5:4_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><a NAME="E6:4_4_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><b>hence </b><a NAME="E7:4_4_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>thus </b><a NAME="E8:4_4_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>thus </b><a NAME="E9:4_4_1_2"/>
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E45:4"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
