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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> , <font color="Maroon">r</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>



<b>thus </b><a NAME="E1:15"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">p</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">r</font> implies <font color="Maroon">p</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:15_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">p</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E2:15_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">q</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:15_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E6:15_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:15_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E9:15_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E10:15_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> holds <br/><font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> =  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T1">FINSEQ_2:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">t</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="polynom3.html#D1" target="_self">POLYNOM3:def 1</a><br/>

<b>thus </b><a NAME="E12:15_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><a NAME="E3:15_1_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><a NAME="E3:15_1_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1_2"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_3"><b>suppose </b></a><a NAME="E1:15_1_1_1_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_1_1_3"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:15_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">t</font>
 ;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E15:15_1"/><i><font color="Green">E71</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E16:15_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E17:15_1"/>
( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E18:15_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">r</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , </i>
;<br/><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:15"/><i><font color="Green">E74</font></i>: 
( ( <font color="Maroon">p</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">r</font> ) or ( <font color="Maroon">p</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">r</font> ) or ( <font color="Maroon">p</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">r</font> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_2"/>
( ( <font color="Maroon">p</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">r</font> ) or ( <font color="Maroon">p</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">r</font> ) or ( <font color="Maroon">p</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">r</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1"><b>suppose </b></a><a NAME="E1:15_2_1"/>
( <font color="Maroon">p</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">r</font> )
 ;<br/><div class="add"><a NAME="E2:15_2_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; ( <font color="Maroon">q</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">r</font> or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:15_2_1"/>
<font color="Maroon">p</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2"><b>suppose </b></a><a NAME="E1:15_2_2"/>
( <font color="Maroon">p</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">r</font> )
 ;<br/><div class="add"><a NAME="E2:15_2_2"/><b>then </b>
( ( <font color="Maroon">p</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">q</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> ) &amp; <font color="Maroon">q</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:15_2_2"/>
<font color="Maroon">p</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_3"><b>suppose </b></a><a NAME="E1:15_2_3"/>
( <font color="Maroon">p</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">r</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_2_3"/>
<font color="Maroon">p</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
