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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1_2"/><i><font color="Green">E25</font></i>: 
(  <a href="idea_1.html#K9" target="_self">MUL_MOD</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> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <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="E2:32_1_2"/><i><font color="Green">E26</font></i>: 
(  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:32_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K7" target="_self">ChangeVal_1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<a NAME="E5:32_1_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:32"><i><font color="Green">E24</font></i></a>, <a class="txt" href="idea_1.html#E4:32_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="idea_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E6:32_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <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
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:32_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K7" target="_self">ChangeVal_1</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<a NAME="E9:32_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:32_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="idea_1.html#E8:32_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="idea_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E10:32_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <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
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:32_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K7" target="_self">ChangeVal_1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<a NAME="E13:32_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="idea_1.html#E2:32_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="idea_1.html#E12:32_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="idea_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E14:32_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <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
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E15:32_1_2"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:32"><i><font color="Green">E24</font></i></a>, <a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="idea_1.html#E9:32_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="idea_1.html#E10:32_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="idea_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E16:32_1_2"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:32"><i><font color="Green">E24</font></i></a>, <a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="idea_1.html#E13:32_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="idea_1.html#E14:32_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="idea_1.html#T5" target="_self">Th5</a></i>;<br/>
 <a href="idea_1.html#K8" target="_self">ChangeVal_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<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</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font> = 
 <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <font color="Maroon">c<sub>2</sub></font>,<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#E1:32_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="idea_1.html#E2:32_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="idea_1.html#E4:32_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="idea_1.html#E8:32_1_2"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
 <a href="idea_1.html#K8" target="_self">ChangeVal_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<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</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="idea_1.html#E4:32_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="idea_1.html#E12:32_1_2"><i><font color="Green">E33</font></i></a></i>
;<br/>
<a NAME="E18:32_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E15:32_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="idea_1.html#E16:32_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="idea_1.html#T21" target="_self">Th21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_2_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:32_1_2_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T20">NAT_1:20</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:32_1_2_2_1_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E18:32_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="idea_1.html#T2" target="_self">Th2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:32_1_2_2_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E7:32_1_2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:32_1_2_2_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="idea_1.html#E6:32_1_2_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E9:32_1_2_2_1_1"/>
<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 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a>, <a class="txt" href="idea_1.html#E4:32_1_2_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2_2_1_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E10:32_1_2_2_1_1"/><b>then </b><i><font color="Green">E42</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 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> or <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 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="idea_1.html#E1:32"><i><font color="Green">E24</font></i></a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><a NAME="E11:32_1_2_2_1_1"/><i><font color="Green">E43</font></i>: 
not <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 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:32_1_2_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E13:32_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="idea_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E13:32_1_2_2_1_1"/><b>then </b>
not <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 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:32_1_2_2_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
not 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="idea_1.html#E10:32_1_2_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="idea_1.html#E11:32_1_2_2_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/><a NAME="E15:32_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E6:32_1_2_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="idea_1.html#E14:32_1_2_2_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E16:32_1_2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:32_1_2_2_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T20">NAT_1:20</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:32_1_2_2_1_2"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E18:32_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="idea_1.html#T2" target="_self">Th2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:32_1_2_2_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E7:32_1_2_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:32_1_2_2_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="idea_1.html#E6:32_1_2_2_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E9:32_1_2_2_1_2"/>
<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 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a>, <a class="txt" href="idea_1.html#E4:32_1_2_2_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2_2_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E10:32_1_2_2_1_2"/><b>then </b><i><font color="Green">E42</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 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> or <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 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="idea_1.html#E1:32"><i><font color="Green">E24</font></i></a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><a NAME="E11:32_1_2_2_1_2"/><i><font color="Green">E43</font></i>: 
not <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 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:32_1_2_2_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E9:32_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2_2_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="idea_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E13:32_1_2_2_1_2"/><b>then </b>
not <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 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:32_1_2_2_1_2"/><b>then </b>
not 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="idea_1.html#E10:32_1_2_2_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="idea_1.html#E11:32_1_2_2_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/><a NAME="E15:32_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E6:32_1_2_2_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E16:32_1_2_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E20:32_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:32_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="idea_1.html#E2:32_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="idea_1.html#E8:32_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="idea_1.html#E12:32_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="idea_1.html#T19" target="_self">Th19</a></i>;<br/>


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