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<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">m</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E33</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> &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; <font color="Maroon">k1</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">k1</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">k1</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">k1</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="E2:46"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">k2</font> <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"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">k2</font> <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"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:46"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k2</font> <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"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:46"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">k2</font> <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"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font>
 ;<br/>

<a NAME="E6:46"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">k2</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>
 
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, , </i>;<br/>
<a NAME="E7:46"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">k2</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="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, , </i>;<br/>
<b>consider </b><font color="Maroon">I1</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="E9:46"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">I1</font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <font color="Maroon">m</font>,<font color="Maroon">k1</font>,<font color="Maroon">n</font>
 ;<br/>
<b>consider </b><font color="Maroon">I2</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="E11:46"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">I2</font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <font color="Maroon">I1</font>,<font color="Maroon">k2</font>,<font color="Maroon">n</font>
 ;<br/>
<a NAME="E12:46"/><i><font color="Green">E46</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E48</font></i>:  <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> = 
 <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">I1</font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E9:46"><i><font color="Green">E44</font></i></a>, </i>
<br/>.= 
 <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">I2</font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E11:46"><i><font color="Green">E45</font></i></a>, </i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E14:46"/>
 <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> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:46"/><b>then </b><i><font color="Green">E49</font></i>: 
4 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:46"/>
3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:46"/><b>then </b>
3 <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E18:46"/><b>then </b><i><font color="Green">E51</font></i>: 
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E19:46"/>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:46"/><b>then </b>
2 <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E21:46"/><b>then </b><i><font color="Green">E53</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:46"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:46"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E24:46"/><b>then </b><i><font color="Green">E55</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </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="E1:46_2"/><i><font color="Green">E56</font></i>: 
<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/><a NAME="E2:46_2"/><b>then </b>
<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">I1</font></span>)</span>
 <b>by </b><i><a class="txt" href="idea_1.html#E9:46"><i><font color="Green">E44</font></i></a>, </i>;<br/><a NAME="E3:46_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:46_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_2_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="46_2_1_1_1"><b>suppose </b></a><a NAME="E1:46_2_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><font color="Maroon">I2</font> <a href="wsierp_1.html#K3">.</a> <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">I1</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E11:46"><i><font color="Green">E45</font></i></a>, <a class="txt" href="idea_1.html#E21:46"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
 <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p1">(<span class="default"><a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E9:46"><i><font color="Green">E44</font></i></a>, <a class="txt" href="idea_1.html#E18:46"><i><font color="Green">E51</font></i></a>, </i>
<br/>.= 
 <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"><a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="idea_1.html#E6:46"><i><font color="Green">E42</font></i></a>, </i>
<br/>.= 
 <a href="idea_1.html#K9" target="_self">MUL_MOD</a> 1,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="idea_1.html#E24:46"><i><font color="Green">E55</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_2"><b>suppose </b></a><a NAME="E1:46_2_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><font color="Maroon">I2</font> <a href="wsierp_1.html#K3">.</a> <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">I1</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E11:46"><i><font color="Green">E45</font></i></a>, <a class="txt" href="idea_1.html#E21:46"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
 <a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p1">(<span class="default"><a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E9:46"><i><font color="Green">E44</font></i></a>, , </i>
<br/>.= 
 <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"><a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
 <a href="idea_1.html#K4" target="_self">ADD_MOD</a> 0,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="idea_1.html#E4:46"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
<font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="idea_1.html#E24:46"><i><font color="Green">E55</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_3"><b>suppose </b></a><a NAME="E1:46_2_1_1_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><font color="Maroon">I2</font> <a href="wsierp_1.html#K3">.</a> <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">I1</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E11:46"><i><font color="Green">E45</font></i></a>, <a class="txt" href="idea_1.html#E21:46"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
 <a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p1">(<span class="default"><a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E9:46"><i><font color="Green">E44</font></i></a>, <a class="txt" href="idea_1.html#E15:46"><i><font color="Green">E49</font></i></a>, </i>
<br/>.= 
 <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"><a href="idea_1.html#K4" target="_self">ADD_MOD</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
 <a href="idea_1.html#K4" target="_self">ADD_MOD</a> 0,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="idea_1.html#E5:46"><i><font color="Green">E40</font></i></a>, </i>
<br/>.= 
<font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="idea_1.html#E24:46"><i><font color="Green">E55</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_4"><b>suppose </b></a><a NAME="E1:46_2_1_1_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><font color="Maroon">I2</font> <a href="wsierp_1.html#K3">.</a> <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">I1</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E11:46"><i><font color="Green">E45</font></i></a>, <a class="txt" href="idea_1.html#E21:46"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
 <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p1">(<span class="default"><a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E9:46"><i><font color="Green">E44</font></i></a>, <a class="txt" href="idea_1.html#E13:46"><i><font color="Green">E48</font></i></a>, </i>
<br/>.= 
 <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"><a href="idea_1.html#K9" target="_self">MUL_MOD</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k2</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="idea_1.html#E7:46"><i><font color="Green">E43</font></i></a>, </i>
<br/>.= 
 <a href="idea_1.html#K9" target="_self">MUL_MOD</a> 1,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="idea_1.html#E24:46"><i><font color="Green">E55</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_5"><b>suppose </b></a><a NAME="E1:46_2_1_1_5"/><i><font color="Green">E97</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>hence </b><font color="Maroon">I2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">I1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E11:46"><i><font color="Green">E45</font></i></a>, <a class="txt" href="idea_1.html#E21:46"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
<font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E9:46"><i><font color="Green">E44</font></i></a>, , <a class="txt" href="idea_1.html#E24:46"><i><font color="Green">E55</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:46_2"/>
<font color="Maroon">I2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E26:46"/>
 <a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <font color="Maroon">m</font>,<font color="Maroon">k1</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k2</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E9:46"><i><font color="Green">E44</font></i></a>, <a class="txt" href="idea_1.html#E11:46"><i><font color="Green">E45</font></i></a>, <a class="txt" href="idea_1.html#E12:46"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


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