<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G</font> holds <br/><span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Olive" title="b1">G</font></span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> implies <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:40"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/>

<a NAME="E2:40"/><span class="kw">then </span>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E3:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:40"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></span>;<br/>
<a NAME="E5:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">A</font> = <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c2">G</font></span>)</span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> ($1,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E8:40"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E9:40"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span>
<a NAME="E10:40"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:40"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
 <a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>)</span>)</span> where <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#T6" title="GOBRD13:th.6">GOBRD13:6</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:40_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_1"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:40_1_1"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E5:40_1_1"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a></span>;<br/>
<a NAME="E7:40_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E4:40_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E8:40_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>;<br/>
<a NAME="E9:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">y</font>,1) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a></span>;<br/>
<font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">y</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E4:40_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E5:40_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a></span>
<br/>.= 
<a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">y</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#D5" title="PSCOMP_1:def.5">PSCOMP_1:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:40_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:40_1"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E6:40_1"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E7:40_1"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="funct_2.html#T65" title="FUNCT_2:th.65">FUNCT_2:65</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">m</font>, <font color="Maroon" title="c9">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E9:40_1"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c9">n</font>)
 <span class="kw">and </span><br/><a NAME="E10:40_1"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c9">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a></span>;<br/>
<a NAME="E11:40_1"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E12:40_1"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E13:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E14:40_1"/><span class="lab"><font color="Green" title="E43">A20</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E12:40_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c7">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="pscomp_1.html#D5" title="PSCOMP_1:def.5">PSCOMP_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">m</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E11:40_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="txt" href="#E12:40_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E13:40_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:40_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:40"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font>, <font color="Maroon" title="c6">m</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> implies ( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">m</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_2"/><span class="lab"><font color="Green" title="E34">A21</font></span>: 
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> )
 <span class="kw">and </span><br/><a NAME="E2:40_2"/><span class="lab"><font color="Green" title="E35">A22</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">m</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a>  )</span><br/>

<a NAME="E3:40_2"/><span class="lab"><font color="Green" title="E36">A23</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">n9</font> = <font color="Maroon" title="c5">n</font>, <font color="Maroon" title="c8">m9</font> = <font color="Maroon" title="c6">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E5:40_2"/><span class="lab"><font color="Green" title="E37">A24</font></span>: 
( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E6:40_2"/><span class="lab"><font color="Green" title="E38">A25</font></span>: 
( <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">n9</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">m9</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E34">A21</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:40_2"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E3:40_2"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="txt" href="#E5:40_2"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> </span><br/>

<a NAME="E8:40_2"/>
<font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">m9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E6:40_2"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E3:40_2"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:40_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_2"><span class="lab"><font color="Green" title="E37">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:40"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="pre_poly.html#T9" title="PRE_POLY:th.9">PRE_POLY:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:40"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E28">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
