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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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="E1:66"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></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="E2:66"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 <b>and </b><br/><a NAME="E3:66"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E4:66"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 1 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 2 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 3 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 4 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 5 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 6 <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E5:66"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 2</span>)</span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 3</span>)</span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 4</span>)</span>,<font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E6:66"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 5 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> 6 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> 6 )
 ;<br/>

<a NAME="E7:66"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K14" target="_self">MESSAGES</a>  &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></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/>
<a NAME="E8:66"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E9:66"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 
<b>by </b><i><a class="txt" href="idea_1.html#E2:66"><i><font color="Green">E51</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E10:66"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></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">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></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">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></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">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></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">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="idea_1.html#E2:66"><i><font color="Green">E51</font></i></a>, <a class="ref" href="idea_1.html#T28" target="_self">Th28</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E8:66"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <span class="p2">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#D21" target="_self">Def21</a></i>
<br/>.= 
 <a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <span class="p1">(<span class="default"><a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <span class="p2">(<span class="default"><a href="idea_1.html#K11" target="_self">IDEAoperationA</a> <span class="p3">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="idea_1.html#D22" target="_self">Def22</a></i>
<br/>.= 
 <a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <span class="p1">(<span class="default"><a href="idea_1.html#K12" target="_self">IDEAoperationB</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:66"><i><font color="Green">E50</font></i></a>, <a class="txt" href="idea_1.html#E4:66"><i><font color="Green">E53</font></i></a>, <a class="txt" href="idea_1.html#E5:66"><i><font color="Green">E54</font></i></a>, <a class="txt" href="idea_1.html#E9:66"><i><font color="Green">E57</font></i></a>, <a class="txt" href="idea_1.html#E10:66"><i><font color="Green">E58</font></i></a>, <a class="ref" href="idea_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:66"><i><font color="Green">E50</font></i></a>, <a class="txt" href="idea_1.html#E2:66"><i><font color="Green">E51</font></i></a>, <a class="txt" href="idea_1.html#E3:66"><i><font color="Green">E52</font></i></a>, <a class="txt" href="idea_1.html#E4:66"><i><font color="Green">E53</font></i></a>, <a class="txt" href="idea_1.html#E6:66"><i><font color="Green">E55</font></i></a>, <a class="ref" href="idea_1.html#T32" target="_self">Th32</a></i>
;<br/>
<b>hence </b><a NAME="E12:66"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K22" target="_self">IDEA_QE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K21" target="_self">IDEA_PE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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