<?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">Key1</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">lk</font>,6, <a href="numbers.html#K5">NAT</a> , <font color="Maroon">Key2</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/>





<a NAME="E1:77"/>
for <font color="Olive">m</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">lk</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 is <a href="int_2.html#V1">prime</a> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 &amp; <font color="Olive">m</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Olive">m</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Olive">m</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Olive">m</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> holds <br/>( <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 ) ) holds <br/><span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Olive">r</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Olive">r</font></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">lk</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 is <a href="int_2.html#V1">prime</a> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 &amp; <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>( <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 ) ) implies <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> );<br/>
<a NAME="E1:77_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:77_1_1"/><i><font color="Green">E35</font></i>: 
 <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">Key1</font>,<font color="Maroon">n</font>,0</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E2:77_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E3:77_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,0</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:77_1_1"/><b>then </b>
 <a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,0</span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,0</span>)</span> = 
 <a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,0
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>
;<br/>
<a NAME="E6:77_1_1"/><b>then </b>
 <a href="funct_7.html#K4">compose</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">Key1</font>,<font color="Maroon">n</font>,0</span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,0</span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a>  <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i><a class="ref" href="funct_7.html#T41">FUNCT_7:41</a></i>;<br/>
<b>hence </b><a NAME="E7:77_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="idea_1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:77_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2"><b>proof </b></a><div class="add">

<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="E1:77_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:77_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">lk</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E3:77_1_2"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 is <a href="int_2.html#V1">prime</a>
 <b>and </b><br/><a NAME="E4:77_1_2"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 <b>and </b><br/><a NAME="E5:77_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E6:77_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 holds <br/>( <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 )
 ;<br/>

<a NAME="E7:77_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <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#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:77_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/>( <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:77_1_2_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:77_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E3:77_1_2_1"/>
( <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,2 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,3</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,3 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,2</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,4 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,4</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,5 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,5 &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,6 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,6 )
 <b>by </b><i><a class="ref" href="idea_1.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:77_1_2"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E10:77_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </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="77_1_2_2"><b>proof </b></a><div class="add">

<i><font color="Green">E55</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> = 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T43">FUNCT_7:43</a></i>
;<br/>
<a NAME="E2:77_1_2_2"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:77_1_2_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E2:77_1_2_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:77_1_2_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</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/>
<b>hence </b><a NAME="E4:77_1_2_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </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="idea_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:77_1_2"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E12:77_1_2"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></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="77_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:77_1_2_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <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/>
<b>hence </b><a NAME="E2:77_1_2_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 <b>by </b><i>, <a class="ref" href="idea_1.html#D2" target="_self">Def2</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:77_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E14:77_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="idea_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E16:77_1_2"/><i><font color="Green">E61</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 &amp; <font color="Maroon">M</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">M</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">M</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">M</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, <a class="txt" href="idea_1.html#E11:77_1_2"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E17:77_1_2"/>
(  <a href="matrix_1.html#K1">width</a> <font color="Maroon">Key1</font> <a href="hidden.html#R1">=</a> 6 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">Key2</font> <a href="hidden.html#R1">=</a> 6 )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, <a class="ref" href="idea_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E18:77_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key1</font></span>)</span> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key1</font></span>)</span> &amp; 3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key1</font></span>)</span> &amp; 4 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key1</font></span>)</span> &amp; 5 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key1</font></span>)</span> &amp; 6 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key1</font></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key2</font></span>)</span> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key2</font></span>)</span> &amp; 3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key2</font></span>)</span> &amp; 4 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key2</font></span>)</span> &amp; 5 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key2</font></span>)</span> &amp; 6 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Key2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E20:77_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>
<a NAME="E21:77_1_2"/>
( <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,1 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,2 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,3 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,4 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,5 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 6 <a href="hidden.html#R1">=</a> <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,6 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,1 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,2 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,3 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,4 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,5 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 6 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,6 )
 
<b>by </b><i><a class="ref" href="idea_1.html#T7" target="_self">Th7</a>, <a class="ref" href="idea_1.html#T8" target="_self">Th8</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E22:77_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 3 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 4 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 5 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 5 &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 6 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> 6 )
 
<b>by </b><i><a class="ref" href="idea_1.html#T4" target="_self">Th4</a>, <a class="ref" href="idea_1.html#T8" target="_self">Th8</a></i>;<br/>
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p4">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="idea_1.html#E9:77_1_2"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="idea_1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_7.html#T50">FUNCT_7:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p4">(<span class="default"><span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="funct_7.html#T50">FUNCT_7:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p4"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p5">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="funct_7.html#T53">FUNCT_7:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" target="_self">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key2</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key1</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p4">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="idea_1.html#T9" target="_self">Th9</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="idea_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="idea_1.html#T1" target="_self">Th1</a>, <a class="ref" href="idea_1.html#T2" target="_self">Th2</a>, , , <a class="ref" href="idea_1.html#T3" target="_self">Th3</a>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="funct_7.html#T50">FUNCT_7:50</a></i>
;<br/>
<b>hence </b><a NAME="E24:77_1_2"/>
<span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:77_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:77"/>
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">lk</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 is <a href="int_2.html#V1">prime</a> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 &amp; <font color="Olive">m</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Olive">m</font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Olive">m</font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Olive">m</font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> holds <br/>( <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,3 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K6" target="_self">NEG_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,2</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K10" target="_self">INV_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,4</span>)</span>,<font color="Maroon">n</font> &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,5 &amp; <font color="Maroon">Key1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 <a href="hidden.html#R1">=</a> <font color="Maroon">Key2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,6 ) ) holds <br/><span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key1</font>,<font color="Maroon">n</font>,<font color="Olive">r</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="idea_1.html#K18" target="_self">IDEA_Q_F</a> <font color="Maroon">Key2</font>,<font color="Maroon">n</font>,<font color="Olive">r</font></span>)</span></span>)</span>,<a href="idea_1.html#K14" target="_self">MESSAGES</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font>
 ;<br/>


</div>
