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

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">lk</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">Key</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">lk</font>,6, <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:71"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:71"/><i><font color="Green">E35</font></i>: 
 <a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font> is   <a href="funct_7.html#NM1">FuncSequence</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:71"/><i><font color="Green">E36</font></i>: 
 <a href="funct_7.html#K7">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<a NAME="E1:71_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:71_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</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="E3:71_1"/><b>then </b><i><font color="Green">E39</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> <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="E4:71_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:71_1"/><b>then </b>
<font color="Maroon">k</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> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:71_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:71_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:71_1"/><b>then </b>
not  <a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<b>then </b> <a href="funct_7.html#K7">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> = 
 <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D7">FUNCT_7:def 7</a></i>
<br/>.= 
 <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_5.html#T21">FUNCT_5:21</a></i>
;<br/>
<b>hence </b><a NAME="E10:71_1"/>
 <a href="funct_7.html#K7">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:71"/>
 <a href="funct_7.html#K6">firstdom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>proof </b></a><div class="add">

<a NAME="E1:71_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:71_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</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="E3:71_2"/><b>then </b><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</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="E4:71_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:71_2"/><b>then </b>
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> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:71_2"/><b>then </b><i><font color="Green">E43</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:71_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:71_2"/><b>then </b>
not  <a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<b>hence </b> <a href="funct_7.html#K6">firstdom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> = 
 <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D6">FUNCT_7:def 6</a></i>
<br/>.= 
 <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key</font>,1</span>)</span>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key</font>,1</span>)</span>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_5.html#T21">FUNCT_5:21</a></i>
<br/>.= 
 <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:71"/>
 <a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font> is    <a href="funct_7.html#M1">FuncSequence</a> of  <a href="idea_1.html#K14" target="_self">MESSAGES</a> , <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 <b>by </b><i>, , <a class="ref" href="funct_7.html#D9">FUNCT_7:def 9</a></i>;<br/>


</div>
