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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 3 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 4 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E1:37_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font></span>)</span> holds <br/> ex <font color="Olive">x</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">j</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:37_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4 or ( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b>
 <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font>
;<br/><b>thus </b><a NAME="E2:37_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>, <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>take </b>
 <a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font>
;<br/><b>thus </b><a NAME="E2:37_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>, <a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>take </b>
 <a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font>
;<br/><b>thus </b><a NAME="E2:37_1_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>, <a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_4"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>take </b>
 <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font>
;<br/><b>thus </b><a NAME="E2:37_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>, <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_5"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
;<br/><b>thus </b><a NAME="E2:37_1_1_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>,<font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:37_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:37_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font></span>)</span>
 <b>and </b><br/><a NAME="E5:37_1_1"/><i><font color="Green">E45</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="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">z</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E6:37_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</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="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</font> holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">z</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font> ) &amp; ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">z</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font> ) &amp; ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 3 implies <font color="Maroon">z</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font> ) &amp; ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 4 implies <font color="Maroon">z</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font> ) &amp; ( <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 implies <font color="Maroon">z</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> ) ) ) )
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


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