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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:92_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="nat_3.html#NK12" target="_self">pfexp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="nat_3.html#K5" target="_self">min</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><a NAME="E3:92_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E4:92_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="int_2.html#T5">INT_2:5</a></i>;<br/><a NAME="E5:92_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E6:92_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:92_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nat_3.html#E4:92_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E7:92_1"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:92_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nat_3.html#E4:92_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E8:92_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_3.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E9:92_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:92_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E10:92_1"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:92_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E11:92_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_3.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E12:92_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:92_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E13:92_1"/><i><font color="Green">E59</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:92_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E14:92_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E15:92_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E16:92_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E6:92_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="nat_3.html#E14:92_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/><a NAME="E17:92_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E6:92_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="nat_3.html#E15:92_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/><b>thus </b><a NAME="E18:92_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K5" target="_self">min</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_1_1"/>
( <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> or <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#K5" target="_self">min</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E8:92_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nat_3.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:92_1_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E10:92_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_3.html#E17:92_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_3.html#T4" target="_self">Th4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_1_1_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:92_1_1_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E9:92_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_3.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:92_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E8:92_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_3.html#E11:92_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="nat_3.html#E1:92_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nat_3.html#E1:92_1_1_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:92_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E12:92_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_3.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E5:92_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E7:92_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E2:92_1_1_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:92_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K5" target="_self">min</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:92_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="nat_3.html#E2:92_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nat_3.html#E3:92_1_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_2"><b>suppose </b></a><a NAME="E1:92_1_1_1_2"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#K5" target="_self">min</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E11:92_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_3.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:92_1_1_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E13:92_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nat_3.html#E16:92_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="nat_3.html#T4" target="_self">Th4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_1_1_1_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:92_1_1_1_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E12:92_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_3.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:92_1_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E8:92_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_3.html#E11:92_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="nat_3.html#E1:92_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nat_3.html#E1:92_1_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:92_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E9:92_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_3.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E5:92_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E7:92_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E2:92_1_1_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:92_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K5" target="_self">min</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:92_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="nat_3.html#E2:92_1_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nat_3.html#E3:92_1_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:92"/>
 <a href="nat_3.html#NK12" target="_self">pfexp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="nat_3.html#K5" target="_self">min</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
