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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> );<br/>
<a NAME="E1:11"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>3</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  holds <br/>( <font color="Olive">b<sub>2</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E2:11_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:11_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a>
 ;<br/>

<b>then 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="E5:11_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pythtrip.html#D3" target="_self">Def3</a></i>;<br/>
<b>assume </b><a NAME="E6:11_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<a NAME="E7:11_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1"><b>suppose </b></a><a NAME="E1:11_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_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="E1:11_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:11_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:11_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="newton.html#T65">NEWTON:65</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:11_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:11_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="newton.html#T65">NEWTON:65</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_2"><b>suppose </b></a><a NAME="E1:11_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> )
 <b>by </b><i><a class="ref" href="nat_1.html#T40">NAT_1:40</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_3"><b>suppose </b></a><a NAME="E1:11_1_1_3"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:11_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green">E7</font></i></a></i>;<br/><a NAME="E3:11_1_1_3"/><b>then </b><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E4:11_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1_1_3"><i><font color="Green">E11</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then 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="E6:11_1_1_3"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="int_2.html#V1">prime</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="pythtrip.html#E6:11_1_1_3"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E8:11_1_1_3"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E9:11_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E10:11_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="real_2.html#T144">REAL_2:144</a></i>;<br/><a NAME="E11:11_1_1_3"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="pythtrip.html#E8:11_1_1_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:11_1_1_3"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E13:11_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E5:11_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pythtrip.html#E6:11_1_1_3"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E14:11_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:11_1_1_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><i><font color="Green">E18</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pythtrip.html#E5:11_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pythtrip.html#E16:11_1_1_3"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>

;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1_3_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pythtrip.html#E6:11_1_1_3"><i><font color="Green">E13</font></i></a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:11_1_1_3_2_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:11_1_1_3_2_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E6:11_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pythtrip.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:11_1_1_3_2_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1_1_3_2_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E5:11_1_1_3_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E17:11_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E6:11_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E7:11_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E8:11_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1_1_3_2_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:11_1_1_3_2_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E11:11_1_1_3_2_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_2_1_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E12:11_1_1_3_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pythtrip.html#E1:11_1_1_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_2_1_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E13:11_1_1_3_2_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E14:11_1_1_3_2_1_1"/><b>then </b>
1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E11:11_1_1_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pythtrip.html#E11:11_1_1_3_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/><a NAME="E15:11_1_1_3_2_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pythtrip.html#E3:11_1_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pythtrip.html#E13:11_1_1_3_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pythtrip.html#E10:11_1_1_3_2_1_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pythtrip.html#E17:11_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_1"><i><font color="Green">E21</font></i></a></i>
;<br/><a NAME="E17:11_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E18:11_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E19:11_1_1_3_2_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="pepin.html#K2">^2</a> 
 ;<br/><a NAME="E20:11_1_1_3_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_2_1_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E21:11_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E22:11_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:11_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="wsierp_1.html#T21">WSIERP_1:21</a></i>;<br/><a NAME="E23:11_1_1_3_2_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><a NAME="E24:11_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a>
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="pythtrip.html#E15:11_1_1_3_2_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pythtrip.html#E19:11_1_1_3_2_1_1"><i><font color="Green">E26</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E26:11_1_1_3_2_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i><a class="ref" href="pythtrip.html#D3" target="_self">Def3</a></i>;<br/><font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pythtrip.html#E26:11_1_1_3_2_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="pepin.html#K2">^2</a> 

;<br/><b>hence </b><a NAME="E28:11_1_1_3_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="pythtrip.html#E15:11_1_1_3_2_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pythtrip.html#E19:11_1_1_3_2_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pythtrip.html#E23:11_1_1_3_2_1_1"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:11_1_1_3_2_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:11_1_1_3_2_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E6:11_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pythtrip.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:11_1_1_3_2_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1_1_3_2_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E5:11_1_1_3_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E17:11_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_2"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E6:11_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E7:11_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</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></i>;<br/><a NAME="E8:11_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1_1_3_2_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:11_1_1_3_2_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E11:11_1_1_3_2_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_2_1_2"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E12:11_1_1_3_2_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pythtrip.html#E1:11_1_1_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_2_1_2"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E13:11_1_1_3_2_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E14:11_1_1_3_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E11:11_1_1_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pythtrip.html#E11:11_1_1_3_2_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/><a NAME="E15:11_1_1_3_2_1_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pythtrip.html#E3:11_1_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pythtrip.html#E13:11_1_1_3_2_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pythtrip.html#E10:11_1_1_3_2_1_2"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pythtrip.html#E17:11_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_2"><i><font color="Green">E21</font></i></a></i>
;<br/><a NAME="E17:11_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E18:11_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E19:11_1_1_3_2_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="pepin.html#K2">^2</a> 
 ;<br/><a NAME="E20:11_1_1_3_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_2_1_2"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E21:11_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E22:11_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:11_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="wsierp_1.html#T21">WSIERP_1:21</a></i>;<br/><a NAME="E23:11_1_1_3_2_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><a NAME="E24:11_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a>
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="pythtrip.html#E15:11_1_1_3_2_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pythtrip.html#E19:11_1_1_3_2_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E26:11_1_1_3_2_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i><a class="ref" href="pythtrip.html#D3" target="_self">Def3</a></i>;<br/><font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_2_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pythtrip.html#E26:11_1_1_3_2_1_2"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="pepin.html#K2">^2</a> 

;<br/><b>hence </b><a NAME="E28:11_1_1_3_2_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="pythtrip.html#E15:11_1_1_3_2_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pythtrip.html#E19:11_1_1_3_2_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pythtrip.html#E23:11_1_1_3_2_1_2"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:11"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>(<a class="txt" href="pythtrip.html#E1:11"><i><font color="Green">E5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:11"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  implies ( <font color="Maroon">c<sub>1</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="pythtrip.html#V1" target="_self">square</a> ) )
 ;<br/>


</div>
